Download The physics of projectiles

North Berwick High School
Department of Physics
Higher Physics
Unit 1 Our Dynamic Universe
Section 4 Gravitation
Section 4
Gravitation
Note Making
Make a dictionary with the meanings of any new words.
The physics of projectiles
1.
State what is meant by a projectile.
Horizontal projectiles
1.
2.
Describe an experiment to confirm that the horizontal and
vertical components of motion are entirely independent of one
another.
Your teacher will show you how to lay out a horizontal projectile
calculation using question 1 on page 24 as an example. Copy
this into your notes.
Projectiles at an angle to the horizontal.
1.
2.
3.
Show how the values for the horizontal and vertical components
of the launch velocities are found.
Copy the summary table on page 8 for any projectile.
Your teacher will show you how to layout an angled projectile
calculation using question 6 on page 26 as an example. Copy
this into your notes.
Projectiles and Newton’s thought experiment
1.
2.
Describe Newton's thought experiment.
Consider the questions on page 11. Are you able to answer any?
(There is no need to make a note.)
The law of universal gravitation
1.
2.
3.
State Newton's theory of universal gravitation.
Copy the formula along with the units.
Copy the examples on pages 17 - 19.
The force of gravity on a planetary scale
1.
Write a note explaining gravity assist. You might find the tennis
ball analogy useful.
The force of gravity on a universal scale
1.
2.
3.
Describe how a star is born within a nebula.
Describe how planets may now be formed around the star.
Consider the same questions again on page 23. Are you able to
answer any now? (Still no need to make a note.) Think about
the next set of questions on page 23.
Section 4
Gravitation
Contents
Content Statements ......................................................................... 1
The physics of projectiles ................................................................. 2
Projectiles and Newton’s thought experiment .................................. 9
Understanding gravity .................................................................... 10
Mapping the gravitational field on Earth ........................................ 12
The Earth’s natural satellite: the Moon .......................................... 14
Evidence in support of the geocentric model .................................. 14
Evidence in support of a new model – heliocentrism ...................... 15
The Moon – what keeps it in place? ............................................... 15
The law of universal gravitation ..................................................... 16
The force of gravity in everyday life ............................................... 17
The force of gravity on a subatomic scale ....................................... 19
The force of gravity on a planetary scale ........................................ 20
The force of gravity on a universal scale ......................................... 21
A huge cloud of dust ...................................................................... 22
A star is born ................................................................................. 22
The formation of the planets .......................................................... 22
Problems ....................................................................................... 24
Solutions ........................................................................................ 29
Content Statements
Content
Notes
Context
a) Projectiles and
Satellites.
Resolving the motion
of a projectile with an
initial velocity into
horizontal and vertical
components and their
use in calculations.
Comparison of
projectiles with
objects in free fall.
Newton’s thought
experiment and an
explanation of why
satellites remain in
orbit.
Using software to
analyse videos of
projectiles
Low orbit and
geostationary satellites.
Satellite communication
and surveying.
Environmental
monitoring of the
conditions of the
atmosphere.
b) Gravity and
mass.
Gravitational Field
Strength of planets,
natural satellites and
stellar objects.
Calculating the force
exerted on objects
placed in a gravity
field.
Newton’s Universal
Law of Gravitation.
Methods for measuring
the gravitational field
strength on Earth. Using
the slingshot effect to
travel in space. Lunar
and planetary orbits.
Formation of the solar
system by the
aggregation of matter.
Stellar formation and
collapse.
The status of our
knowledge of gravity as
a force may be
explored. The other
fundamental forces
have been linked but
there is as yet no
unifying theory to link
them to gravity.
1
Section 4 Gravitation
The physics of projectiles
© Gustoimages/Science Photo Library
A projectile is an object on which the only force acting is gravity. There are a
variety of examples of projectiles: an object dropped from rest is a projectile
(provided that the influence of air resistance is negligible), an object thrown
vertically upwards is a projectile (provided that the influence of air resistance is
negligible) and an object thrown upwards at an angle is also a projectile (the
same assumption). A projectile is any object, which, once projected, continues
its motion by its own inertia and is influenced only by the downward force of
gravity.
By definition, a projectile has only one force acting on it: the force of gravity. If
there were any other force acting on an object, then that object would not be a
2
projectile. Projectiles can be launched horizontally, vertically or at an angle
between the two.
All projectiles have a horizontal and vertical velocity.
They also have a horizontal and vertical displacement.
The equations of motion can be used to describe the motion of projectiles. You
have already investigated one type of projectile: a vertical projectile. This is the
case of the tennis ball allowed to fall freely under the force of gravity, or the
tennis ball allowed to fall freely under the force of gravity and allowed to
bounce, or the tennis ball thrown up in the air and allowed to return to its
starting position.
To summarise, for a vertical projectile:
Direction of
motion
Forces
Velocity
Acceleration
Constant (in this
case 0 m s –1 )
None
Horizontal
Air resistance
negligible so
no forces in
the horizontal
Changing with
time
Vertical
Air resistance
negligible so
only force of
gravity acting
in the vertical
Constant or
uniform
acceleration
(– 9.8 m s -2 )
3
The second projectile situation to consider is the horizontal projectile.
Picture a motorcyclist on the top of a tall building
about to perform a death-defying stunt of
incredible skill. Do not try this at home! Predict her
as she drives across the roof at speed and off the
edge..
Explain your prediction.
path
You should investigate horizontal projectiles to confirm that the horizontal and
vertical components of motion are entirely independent of one another. This
will also give you an opportunity to verify the equations of motion, i.e. to
confirm that the equations we use correctly describe the motion that happens
in real life.
Some of the experiments you might experience include:
filming motion of a horizontal projectile and confirming the horizontal
and vertical motion using software such as tracker.jar
using a traditional piece of equipment that allows you to compare the
time of flight for an object dropped vertically and an object fired
horizontally, both being released simultaneously and from the same
height
4
using a thought experiment and a spreadsheet programme to calculate
and graph horizontal displacement against vertical displacement for an
object dropped vertically and an object fired horizontally, both being
released simultaneously and from the same height
using a ballistics cart, e.g. PASCO
using a projectile launcher and timing system to confirm that the time of
flight recorded matches that which can be determined theoretically,
estimating and accounting for experimental uncertainties.
To summarise, for a horizontal projectile:
Direction of
motion
Forces
Velocity
Acceleration
Constant
None
Horizontal
Air resistance
negligible so
no forces in
the horizontal
Changing with
time
Vertical
Air resistance
negligible so
only force of
gravity acting
in the vertical
Constant or
uniform
acceleration
(– 9.8 m s -2 )
5
The third and final projectile situation to consider is the projectile at an angle
to the horizontal.
© Gustoimages/Science Photo Library
Remember that any vector can be resolved into its horizontal and vertical
components:
launch velocity (ms -1 )
θ
cos θ
=
adjacent
hyp
adjacent
=
hyp cos θ
horizontal component
=
launch velocity cos θ
6
launch velocity (ms -1 )
θ
sin θ
=
opposite
hyp
opposite
=
hyp sin θ
=
launch velocity sin θ
vertical component
The distance travelled horizontally (the range) is determined by the cosine
component of the launch velocity. The time of flight is determined by the sine
component of the launch velocity.
Determine experimentally and/or theoretically the angle, which gives the
greatest range for a fixed, launch velocity.
Your teacher will show you how to lay out the answer to question 1 on page 24.
7
To summarise, for a projectile at an angle to the horizontal:
Direction of
motion
Forces
Velocity
Acceleration
Constant
None
Horizontal
Air resistance
negligible so
no forces in
the horizontal
Changing with
time
Vertical
Air resistance
negligible so
only force of
gravity acting
in the vertical
Constant or
uniform
acceleration
(– 9.8 m s -2 )
Forces
Velocity
Acceleration
Constant
None
Horizontal
Air resistance
negligible so
no forces in
the horizontal
Changing with
time
Vertical
Air resistance
negligible so
only force of
gravity acting
in the vertical
Constant or
uniform
acceleration
(– 9.8 m s -2 )
To summarise, for any projectile:
Direction of
motion
8
Projectiles and Newton’s thought experiment
Think back to our death-defying motorcyclist. Consider what would happen if:
the building were taller
the horizontal launch velocity were greater
the Earth curved away more steeply.
This is what Newton (1643–1727) thought about. He considered that there is a
horizontal launch velocity that would result in the projectile continually falling
to Earth as a result of the force of gravity at the same rate at which the Earth’s
surface curves away beneath it. The projectile would continually fall but would
not hit the surface. It would therefore orbit the Earth. His thought experiment
explained satellite motion around 300 years before it became a reality.
On October 4 1957 Newton’s thought experiment became reality with the
successful launch of the world’s first artificial satellite, Sputnik. The significance
of this launch in terms of its impact on life cannot be underestimated. Low -orbit
and geostationary satellites are used for a wide ran ge of applications, including
environmental monitoring, military applications and defence, communications,
and scientific investigations.
9
Understanding gravity
Sir Isaac Newton (1642–1727) did not discover gravity. People had long been
aware that objects when released would fall.
www.Cartoonstock.com
Galileo (1564–1642) had carried out experiments on falling objects and put
forward a theory on the motion of objects in freefall in his unfinished work De
Motu (On Motion).
Obviously, then, Galileo was performing experiments at the very beginning of
his investigations into motion, and he took his experimental results seriously.
Over two decades he changed his ideas and refined his experiments, and in the
end he arrived at the law of falling bodies which states that in a vacuum all
bodies, regardless of their weight, shape, or specific gravity, are uniformly
accelerated in exactly the same way, and that the distance fallen is proportional
to the square of the elapsed time.
The Galileo Project: http://galileo.rice.edu/sci/theories/on_motion.html
10
Whilst not the first to investigate gravity,Newton’s significant contribution,
ranked ‘among humanity’s greatest achievements in abstract thought’
(http://www.newton.ac.uk/newtlife.html), was to theorise that the force acting
locally on an apple could be applied to the universe. Newton developed the
universal theory of gravitation. This was published in his best -known work, the
Principia.
http://www.newtonprject.sussex.ac.uk
Questions to consider before you progress through this section:
What is gravity?
What is the force of gravity?
What are the effects of gravity?
What do we know about gravity?
How can we make use of gravity?
11
Mapping the gravitational field on Earth
On 17 March 2009, the European Space Agency launched the Gravity Field and
Steady State Ocean Circulation Explorer (GOCE), the lowest orbit research
satellite in operation (at 254.9 km).
© ESA
(see http://www.esa.int/esaLP/ESAYEK1VMOC_LPgoce_0.html ).
The mission of this satellite is to map the Earth’s gravitational field in greater
detail than has previously been possible. This data will be used to:
inform predictions of climate
understand and monitor the effects of climate change, making
accurate measurements of ocean circulation and sea level
develop an improved understand of the internal geology of the Earth,
including hazards such as volcanoes
develop a system that allows comparison of heights all over the world
to be made for the first time; such a system would be very useful
when undertaking large-scale engineering projects such as bridge
building and tunnel construction.
12
© ESA
In order to achieve its very challenging mission objectives, the satellite was
designed to orbit at a very low altitude, where the gravitational variations are
stronger closer to Earth.
Since mid-September 2009, GOCE has been in its gravity-mapping orbit at a
mere 254.9 km mean altitude – the lowest orbit sustained over a long period by
any Earth observation satellite.
The residual air at this low altitude causes the orbit of a standard satellite to
decay very rapidly. GOCE, however, continuously nullifies the drag in real time
by firing an ion thruster using xenon gas.
It ensures the gravity sensors are flying as though they are in pure freefall, so
they pick up only gravity readings and not the disturbing effects from other
forces.
To obtain clean gravity readings, there can be no disturbances from moving
parts, so the entire satellite is a single extremely sensitive measuring device.
13
‘The gravity measuring system is functioning extremely well. The system is
actively compensating for the effects of atmospheric drag and delivering a
continuous set of clean gravity readings,’ Dr Floberghagen a Physicist working
on it said.
‘This in itself is an excellent technical achievement. GOCE has proven to be a
nearly perfect satellite for measuring gravity from space.’
(Extract from http://www.esa.int/SPECIALS/GOCE/SEMY0FOZVAG_0.html)
The ‘standard’ acceleration due to gravity at the Earth’s surface is 9.8 m s –2 .
The GOCE mission has established that the figure in fact varies from 9.788 m s –2
at the equator to 9.838 m s –2 at the poles.
The Earth’s natural satellite: the Moon
Greek philosophers understood that the Moon is a sphere in orbit around the
Earth. They also realised that the Moon is not a light source, but reflects
sunlight.
Around 1850 years ago, Ptolemy (90–168) hypothesised that the Moon and Sun
both orbit the Earth. This theory was the most commonly held belief in ancient
Greece, Europe and in many other parts of the world. There is evidence that
others, including Muslim scholars, developed an alternative theory, one in
which the Earth was not orbited by the Sun, 250 years before Ptolemy proposed
his theory.
Evidence in support of the geocentric model
Geocentrism is the belief that the Earth is the centre of the universe. What
evidence supported this theory? What evidence do we now have to s upport an
alternative theory?
Certainly, the Sun, the Moon, the stars and other planets appear to revolve
around us. In fact, we still commonly talk about the Sun in these terms: ‘rising’,
‘setting’, and ‘going down’. These suggest a belief that the Sun is orbiting the
Earth.
The Ancient Greeks also believed that the Earth is stationary. How do you know
it isn’t?
14
Evidence in support of a new model – heliocentrism
With the work of Copernicus (1473–1543) the prevailing view of the universe
began to change. Kepler (1571–1630) developed three laws that predicted that
the orbits of the planets are elliptical, with the Sun at a focus of the ellipse.
What evidence was there to support this model at the time when Copernicus
and Kepler were developing these theories and laws? What evidence i s there to
support this model now?
The Moon – what keeps it in place?
The Moon remains in orbit around the Earth as a result of the force of gravity. It
is the weakest of the four fundamental forces (the others being the strong
force, the weak force and the electromagnetic force) yet it keeps the universe in
shape!
Newton developed the theory of universal gravitation. This was a very
important piece of work – not least because he proposed it to be universal, i.e.
he proposed that all parts of the universe obey the same laws of nature.
His theory hypothesised (and proved mathematically) that the Moon orbits the
Earth as a result of the same force that causes an apple to fall from a tree. The
moon falls around the Earth.
What evidence did Newton have available to him to support this theory? What
evidence do we now have?
As we have gathered more and more scientific evidence about the universe, we
have come to take for granted the universality of the laws of nature.
15
The law of universal gravitation
Newton’s law of universal gravitation proposed that each body with mass will
exert a force on each other body with mass. The theory states that the force of
gravitational attraction is dependent on the masses of both objects and is
inversely proportional to the square of the distance that separates them.
But Newton remained uncertain. He could not account for action at a distance
without some medium, i.e. he was concerned about the distance over which he
proposed gravitational force would act in space and the fact that space is a
vacuum.
F
Gm1m2
r2
F is force in newtons (N)
m 1 and m 2 are the two masses measured in kilograms (kg)
r is the distance between them (m)
G is the gravitational constant N m 2 kg –2
The value of the gravitational constant was determined by Cavendish (1731 –
1810) in the late 1700s.
It was another hundred years before Boys (1855–1944) improved on its
accuracy.
G = 6.67428 × 10 –11 N m 2 kg –2
G remains one of the most difficult constants to measure with accuracy. In 2007
a further value was published which suggested an improvement on the
accuracy:
G = 6.67 × 10 –11 N m 2 kg –2 is the value we will use for calculations in Higher
Physics.
16
The force of gravity in everyday life
Newton’s law of universal gravitation allows us to calculate the force of gravity
between point masses or spherical objects.
Consider two objects, a folder of mass 0.3 kg and a pen of mass 0.05 kg, placed
on a desk, 0.25 m apart.
Calculate the magnitude of the gravitational force between the two masses
(noting that the force is always attractive). Assume that the objects can be
approximated to spherical objects.
F =?
m 1 = 0.30 kg
m 2 = 0.05 kg
r = 0.25 m
G = 6.67 × 10 –11 N m 2 kg –2
F
Gm1m2
r2
F
6.67 10 –11 0.30 0.05
0.252
F 1.60 10
11
N
The gravitational force always acts in a straight line between the two objects
being considered. Note that the gravitational force is always attractive, unlike
electrostatic and magnetic forces.
17
Consider the gravitational force due to the Earth acting on the pen.
Assume that the Earth and pen can be approximated to spherical objects.
F=?
F
F
F
m 1 = 5.97 × 10 24 kg
m 2 = 0.05 kg
G = 6.67 × 10 –11 N m 2 kg –2
r = 6.38 × 10 6 m
Gm1m2
r2
6.67 10 –11 5.97 1024 0.05
(6.38 106 ) 2
0.489N
How else could this calculation have been carried out?
W = mg
W = 0.05 × 9.8
W = 0.49 N
18
The force of gravity on a subatomic scale
Consider the attractive gravitational force between two protons in a nucleus.
F
F=
Gm1m2
r2
F is force in newtons (N) = ?
m 1 = 1.67 × 10 –27 kg (mass of proton)
m 2 = 1.67 × 10 –27 kg (mass of proton)
r = 0.84 × 10 –15 m (radius of proton; radius of neutron assumed equal)
G = 6.67 × 10 –11 N m 2 kg –2
6.67 × 10 –11
x 1.67 × 10 –27 x
(0.84 × 10 –15 ) 2
1.67 × 10 –27
F = 2.64 × 10 –34 N
The force of gravity is clearly insignificant except where we are dealing with
very large mass. At the subatomic scale, and at very short range, the strong
force is the most significant. The electromagnetic (repulsive) force between two
protons is 3.27 x 10 -16 N. However, although the weakest of the four forces, the
gravitational force acts over enormous distances and on a universal scale!
19
The force of gravity on a planetary scale
The theory of universal gravitation can be used beyond satellite motion. It is
commonly used in space travel in a technique called gravity assist. The gravity
assist technique is often called, incorrectly, the ‘slingshot’ effect. The physics of
the slingshot effect are considerably more complex.
Gravity assist was made famous in the movie ‘Apollo 13’. The craft had been
irreparably damaged by an explosion on board. The planned Moon landing was
abandoned and the priority became the safe return of the astro nauts to Earth.
Gravity assist was the technique by which relatively small amounts of the
limited available fuel could be used to manoeuvre the craft onto a trajectory
that would allow the Moon’s gravitational field to turn the ship. Detailed
information on the mission and the problems encountered is available from
NASA (see http://science.ksc.nasa.gov/history/apollo/apollo-13/apollo13.html).
Gravity assist is routinely used to boost space flight on unmanned missions to
distant planets such as Jupiter and Venus. To understand the principles of this,
let us first consider a simple analogy.
Consider a tennis ball travelling towards a tennis player holding her racquet at
the ready. In this analogy, the ball is the spacecraft and the racquet is a massive
planet. The racquet hits the ball. We will consider the outcome to be that the
player successfully hits the ball, causing it to change direction and increase
speed.
Some questions (and answers):
What happens to the ball? (it changes direction and increases speed)
What type of interaction is this? (a collision)
What quantity is conserved in all collisions? (momentum)
What other quantity is considered in collisions? (energy )
In this case, what can we say about the momentum of the tennis ball? (it
increases in magnitude)
And the momentum of the racquet? (the law of conservation of momentum tells
us it must decrease in magnitude)
What about the energy of the tennis ball? (it increases)
What is the observable effect of the change in momentum of the tennis ball?
(an observable change in speed)
20
Is there an observable effect of the change in momentum of the racquet? (no –
because the racquet has a much larger mass than the ball)
Is there an effect of the change in momentum of the racquet? (yes – it slows
down)
In this case we have considered a mechanical interaction. The gravity -assist
method is a gravitational exchange between the planet and the spacecraft. The
spacecraft, which has very small mass, is able to gain momentum and energy
from the planet, which has enormous mass. The mass difference is such that
whilst the effect on the spacecraft (i.e. an increase in momentum and therefore
speed) is observable and useable, the effect on the planet (i.e. a decrease in
momentum and therefore speed) is not observable and in fact is negligible.
Gravity assist makes use of the universal law of gravitation to enable space
flight with minimal fuel requirements. This reduces the weight of the craft and
allows us to reach greater distance than would otherwise be possible.
The force of gravity on a universal scale
We have explored the effects of the force of gravity on a small scale, its
importance in satellite motion and its use in space flight. We have discussed
some of the historical story associated with our understanding of gravitational
force, but we have yet to discuss a very significant impact of gravitational force.
Much of the current scientific thinking is that gravitation al force is responsible
for the formation of the solar system by aggregation (or accretion) of matter.
Gravitational force also plays a crucial role in the birth and death of stars.
The solar system formed around 4.5 billion years ago from a huge swirlin g cloud
of dust. We know this because advances in technology, such as the Hubble
telescope, have allowed us to look deep into space to observe the birth of stars
similar to our sun.
21
A huge cloud of dust
Throughout the Milky Way, and other galaxies
are gigantic swirling clouds of dust and gas
known as nebulae. It is within nebulae that
are born. Our star, the Sun, was created in one
nebula.
Something, perhaps the shock wave from an
exploding supernova (dying star), triggered
particles to be drawn together to form a dense
spherical cloud. The accumulation of dust set
chain reaction. As the core of the cloud
attracted more dust, its gravitational pull
increased. More and more dust was sucked in
the cloud collapsed in on itself. As this
happened, the rotation of the cloud increased
speed, as happens when spinning ice-skaters
their arms. The rotational forces at the equator
cloud prevented dust along this plane being
in, causing the cloud to flatten into a disc
spinning around a dense core.
like it,
stars
such
dust
off a
and
A pillar of dust and
gas in the Orion
Nebula. © NASA
in
pull in
of the
drawn
A star is born
As more and more mass is accumulated at the centre of the disc, the
temperature increased dramatically. Eventually there was enough energy to set
off nuclear reactions. Hydrogen atoms fused to form helium, releasing
enormous amounts of energy in vigorous bursts. This marked the birth of the
Sun, although it would take between 1 and 10 million more years for it to settle
into the main sequence star recognisable today.
The formation of the planets
The planets, and other extraterrestrial objects such as asteroids, formed in the
flat plane of the spinning disc of dust. Electrostatic forces or sticky carbon
coatings made dust particles stick together to form clusters, which in turn stuck
together to form rocks. Mutual gravity caused these rocks to come together,
eventually to form planets. This ‘coming together’ of material is a process
known as accretion (or aggregation).
22
Let us now return to the questions to consider:
What is gravity?
What is the force of gravity?
What are the effects of gravity?
What do we know about gravity?
How can we make use of gravity?
Are we now in a position to answer these questions conclusively? The answer of
course is no. We have explored the force of gravity and some uses of it.
However, 350 years after Newton identified gravitational force as a ‘universal’
force; it remains the least understood of the four fundamental forces. We have
evidence of its role in the formation of the solar system, but do not conclusively
know that this theory is the correct one. We know that it is associated with
mass but we do not understand the mechanism for this.
In fact there are many questions still unanswered and more research and
experimentation to be done.
What effect does Gravity have on space-time?
Is the graviton the exchange particle of gravitational fields?
Is G, the gravitational constant, really constant and is it really
universal?
What is the nature of mass?
What are Black holes (and white holes?).
What is the universe made of?
Why does the gravitational mass of galaxy exceed the mass of the
known matter? Is there something else there? Or do we not really
understand gravity at all?
When were first stars formed? What were they like?
Do gravitational waves exist?
Why do the four fundamental forces have the strengths they have?
Is there a single unifying theory which links gravitational forc e to the
other much better understood forces?
23
Gravitation Problems
Projectiles (Revision from Section 1)
1.
A plane is travelling with a horizontal velocity of 350 m s
300 m. A box is dropped from the plane.
The effects of friction can be ignored.
(a)
(b)
(c)
2.
Calculate the time taken for the box to reach the ground.
Calculate the horizontal distance between the point
where the box is dropped and the point where it hits the
ground.
What is the position of the plane relative to the box when
the box hits the ground?
Calculate the time of flight of the projectile.
What is the height of the starting point of the projectile
above sea level?
State any assumptions you have made.
A ball is thrown horizontally with a speed of 15 m s 1 from the top of a
vertical cliff. It reaches the horizontal ground at a distance of 45 m from
the foot of the cliff.
(a)
(b)
4.
at a height of
A projectile is fired horizontally with a speed of 12·0 m s 1 from the edge of
a cliff. The projectile hits the sea at a point 60·0 m from the base of the
cliff.
(a)
(b)
3.
1
(i)
Draw a graph of vertical speed against time for the
ball for the time from when it is thrown until it
hits the ground.
(ii)
Draw a graph of horizontal speed against time for
the ball.
Calculate the velocity of the ball 2 s after it is thrown.
(Magnitude and direction are required.)
A football is kicked up at an angle of 70° above the horizontal at
15 m s 1 . Calculate:
(a)
(b)
the horizontal component of the velocity
the vertical component of the velocity.
24
5.
A projectile is fired across level ground and takes 6 s to travel from A to B.
The highest point reached is C. Air resistance is negligible.
C
A
B
Velocity-time graphs for the flight are shown below. V H is the horizontal
velocity and V V is the vertical velocity.
40
V H /m s
0
-1
30
3
6
0
V V /m s
-1
3
6
time /s
time /s
-30
(a)
(b)
(c)
(d)
(e)
Describe:
(i)
the horizontal motion of the projectile
(ii)
the vertical motion of the projectile.
Use a vector diagram to find the speed and angle at
which the projectile was fired from point A.
Find the speed at position C. Explain why this is the
smallest speed of the projectile.
Calculate the height above the ground of point C.
Find the horizontal range of the projectile.
25
6.
A ball of mass 5·0 kg is thrown with a velocity of 40 m s
to the horizontal.
1
at an angle of 30°
40 ms -1
30°
Calculate:
(a)
(b)
(c)
(d)
7.
the
the
the
the
vertical component of the initial velocity of the ball
maximum vertical height reached by the ball
time of flight for the whole trajectory
horizontal range of the ball.
A launcher is used to fire a ball with a velocity of 100 m s 1 at an angle of
60° to the ground. The ball strikes a target on a hill as shown.
100 ms -1
60°
(a)
(b)
Calculate the time taken for the ball to reach the target.
What is the height of the target above the launcher?
26
8.
A stunt driver attempts to jump across a canal of width 10 m.
The vertical drop to the other side is 2 m as shown.
(a)
(b)
(c)
9.
A ball is thrown horizontally from a cliff. The effect of friction can be
ignored.
(a)
(b)
10.
Calculate the minimum horizontal speed required so that
the car reaches the other side.
Explain why your answer to (a) is the minimum horizontal
speed required.
State any assumptions you have made.
Is there any time when the velocity of the ball is parallel
to its acceleration? Justify your answer.
Is there any time when the velocity of the ball is
perpendicular to its acceleration? Justify your answer.
A small ball of mass 0·3 kg is projected at an angle of 60 ° to the horizontal.
The initial speed of the ball is 20 m s 1 .
2020mms
s - -1
1
60 o
Show that the maximum gain in potential energy of the ball is 45 J.
11.
A ball is thrown horizontally with a speed of 20 m s 1 from a cliff. The
effects of air resistance can be ignored. How long after being thrown will
the velocity of the ball be at an angle of 45° to the horizontal?
27
Gravity and mass
In the following questions, when required, use the following data:
Gravitational constant = 6·67 × 10
11
N m 2 kg
2
1.
State the inverse square law of gravitation.
2.
Show that the force of attraction between two large ships, each of mass
5·00 × 10 7 kg and separated by a distance of 20 m,
is 417 N.
3.
Calculate the gravitational force between two cars parked 0·50 m apart.
The mass of each car is 1000 kg.
4.
In a hydrogen atom an electron orbits a proton with a radius of 5·30 ×
10 11 m. The mass of an electron is 9·11 × 10 31 kg and the mass of a
proton is 1·67 × 10 27 kg. Calculate the gravitational force of attraction
between the proton and the electron in a hydrogen atom.
5.
The distance between the Earth and the Sun is 1·50 × 10 11 m. The mass of
the Earth is 5·98 × 10 24 kg and the mass of the Sun is 1·99 × 10 30 kg.
Calculate the gravitational force between the Earth and the Sun.
6.
Two protons exert a gravitational force of 1·16 × 10 35 N on each other.
The mass of a proton is 1·67 × 10 27 kg. Calculate the distance separating
the protons.
28
Solutions
Projectiles
1.
(a)
(b)
7·8 s
2730 m
2.
(a)
(b)
5·0 s
123 m
3.
(b)
24·7 m s
4.
(a)
v horiz = 5·1 ms 1 ,
5.
(b)
(c)
(d)
(e)
50 m s
40 m s
46 m
240 m
1
6.
(a)
(b)
(c)
(d)
20 m s
20 m
4s
140 m
1
7.
(a)
(b)
8s
379 m
8.
(a)
15·6 m s
11.
2s
1
at an angle of 33° below the horizontal
b)
v vert = 14·1 m s
1
at 36.9° above the horizontal
1
1
Gravity and mass
1.
Gm1m 2
r2
F=
3.
2·67 × 10
4
4.
3·61 × 10
47
N
N
5.
3·53 × 10 22 N
6.
4·00 × 10
15
m
29