Download velocity - Uplift Education

EVERYTHING MOVES!!!!! THERE IS NO
APSOLUTE REST!!!!
Even while sitting in the classroom appearing motionless, you
are moving very fast.
• 0.4 km/s (0.25 mi/s) rotating around the center of the Earth
• 30 km/s relative to the Sun
• even faster relative to the center of our galaxy (The Sun
orbits the center of the Milky Way at about 250 km/s and
it takes about 220 million years to complete an orbit. )
• together with the whole galaxy moving away from the center of
the Universe at huge speed. Measurements, confirmed by the
Cosmic Background Explorer satellite in 1989 and 1990, suggest
that our galaxy and its neighbors, are moving at 600 km/s
(1.34 million mi/h) in the direction of the constellation Hydra.).
The Sea
Serpent
When we discuss the motion of something, we describe its
motion relative to something else.
When you say that you drove your car at speed of 50 mi/h, of course
you mean relative to the road or with respect to the surface of the
Earth. You are actually using coordinate system without knowing it.
A frame of reference is a perspective from which a
system is observed together with a coordinate system
used to describe motion of that system.
One of the important problem problems in Physics is this:
if a any given instant in time we know the positions and velocities
of all the particles that make up a particular system can we predict
the future position and velocities of all the particles?
If we can do it then we can:
predict solar eclipses, put satellites into orbit, find out how the
position of a swing varies with time and find out where a soccer
ball ends up when struck by a foot.
Classical Mechanics: - Study of the motion of macroscopic objects and
related concepts of force and energy
Kinematics – is concerned with the description of how objects
move; their motion is described in terms of displacement,
velocity, and acceleration
Dynamics – explains why objects change the state of the motion
(velocity) as they do; explains motion and causes of changes using
concepts of force and energy.
The movement of an object through space can be quite complex.
There can be internal motions, rotations, vibrations, etc…
This is the combination of rotation (around its
center of mass) and the motion along a line parabola.
If we treat the hammer as a particle the only
motion is translational motion (along a line)
through space.
Kinematics in One Dimension
Our objects will be represented as point objects (particles) so
they move through space without rotation.
We’ll be neglecting all factors such as the shape, size, etc.
that make the problem too difficult for now.
Simplest motion: motion of a particle along a line – called:
translational motion or one-dimensional (1-D) motion.
Displacement
is the shortest distance in a given direction.
 The displacement tells us how far an object is from
its starting position and in what direction
P
(it tells us how the object is displaced)
Q
path;
length is distance traveled
Distance between
final and initial point
Example:
1)
x1 = 7 m, x2 = 16 m
2) x1 = 7 m, x2 = 2 m
x3 = 12 m
∆x = 5 m
“+” direction Representation of
displacement in a
∆x =-5m
coordinate system
“ – ” direction
Example:
A racing car travels round a circular track of radius 100 m.
The car starts at O. When it has travelled to P its
displacement as measured from O is
A 100 m due East
B 100 m due West
C 100 √2 m South East
D 100 √2 m South West
Vectors and Scalars
Each physical quantity will be either a scalar or a vector.
Scalar is a quantity that is completely specified by a positive or
negative number with appropriate units.
Temperature, length, mass, time, speed, …
Vector must be specified by both magnitude
(number and unit) and direction.
Displacement, velocity, force,…
Scalar
Vector
distance - 50 km
displacement: 50 km, E
speed - 70 km s-1
velocity:
70 km s-1, S-W
Scalars obey the rules of ordinary algebra:
2 kg of potato + 2 kg of potato = 4 kg of potato
Vectors obey the rules of vectors’ algebra:
The sum of two vectors depends on their directions.
Average and Instantaneous Velocity
DEF: Average velocity is the displacement covered per unit time.
v avg
displacement
=
time
x 2 -x1
Δx
v avg =
=
t 2 -t1
Δt
(it obviously has direction,
the same as displacement)
SI unit : m/s
Instantaneous velocity is the velocity at one instant.
The speedometer of a car reveals information about the instantaneous
speed of your car. It shows your speed at a particular instant in time.
If direction is included you have instantaneous velocity.
Average and Instantaneous Speed
How fast do your eyelids move when you blink? Displacement is zero, so vavg
= 0. How fast do you drive in one hour if you drive zigzag and the magnitude
of the displacement is different from distance?
To get the answers to these questions we introduce speed:
Speed is the distance object covers per unit time.
distance traveled
v=
Δt
it tells us how fast
the object is moving
KCR train has travelled a distance of about 6.7 km from
University Station to Tai Po Market Station. But if we
measure their separation by drawing a straight line on
the map, we will find that Tai Po Market Station is only
5.4 km from University Station, roughly in the NorthWest direction. This is the displacement of the train.
We take the KCR trip from University Station to Tai Po
Market Station. It takes about 6 minutes to travel a distance
of 6.7 km. Thus,
The displacement of Tai Po Market Station from University Station is 5.4 km, so
in the North-West direction. This is smaller than the average speed of the train.
So why do we care of velocity at all? OK, it gives us direction what is very important
(just imagine airplane controller with information only on speed of airplanes not on
directions). But we saw that average speed is greater in general than magnitude of
average velocity. So why is concept of velocity so important?
Because acceleration is a vector, all of equations are vector equations.
Acceleration can be in any direction to the velocity and the motion will depend on that.
ONLY:
if motion is 1-D without changing direction;
speed = magnitude of velocity because
distance traveled = magnitude of displacement
instantaneous speed = magnitude of instantaneous velocity
Example:
A racing car travels round a circular track of radius 100 m.
The car starts at O.
It travels from O to P in 20 s.
Its velocity was 10 m/s.
Its speed was πr/t = 16 m/s.
The car starts at O. It travels from O back to O in 40 s.
Its velocity was 0 m/s.
Its speed was
2πr/t = 16 m/s.
Let’s look at the motion with constant velocity
so called uniform motion
in that case, velocity is the same at all times so
v = vavg at all times, therefore:
v=
x
t
or
x = vt
This is the only equation that we can use for
the motion with constant velocity.
Object moving at constant velocity covers the
same distance in the same interval of time.
click
Acceleration
Acceleration is the change in velocity per unit time.
Δv
a=
Δt
(Change in velocity ÷ time taken)
vector quantity – direction of the change in velocity
In the SI system the unit is
meters per second per second.
m
 a  = s = m/s2
s
a = 3 m/s2 means that velocity changes 3 m/s every second!!!!!!
If an object’s initial velocity is 4 m/s then after one second it will be
7 m/s, after two seconds 10 m/s, ….
Let’s look at the motion with constant acceleration
so called uniformlly accelerated motion
let:
t
a
u
v
x
=
=
=
=
=
the time for which the body accelerates
acceleration
the velocity at time t = 0, the initial velocity
the velocity after time t, the final velocity
the displacement covered in time t
from definition of a:
v-u
a=
t
velocity v at any time t = initial velocity u
increased by a, every second
→ v = u + at
example:
u = 2 m/s
a = 3 m/s2
speed increases 3 m/s
EVERY second.
t(s)
v (m/s)
0
2
1
5
2
8
3
11
v avg =
(2+5+8+11)m/s
= 6.5m/s
4
arithmetic sequence, so
v avg =
(2+11)m/s
= 6.5m/s
2
In general:
u+v
v
=
for the motion with constant acceleration: avg
2
Till now we had three formulas
From definition of average velocity we can find displacement in any case:
x = vavg t
For motion with constant acceleration, velocity changes according:
v = u + at
u+v
and average velocity is: v avg =
2
These three equations are enough to solve any
problem in motion with constant acceleration.
But we are lazy and we want to have more equations that are
nothing new, but only manipulations of this three.
v = u + at
v
u+v
u + u + at
2u at
a 2
x = v avgt =
t=
t=
t+ t = ut+ t
2
2
2
2
2
t=
v-u
a
u+v
v + u  v - u  v 2 - u2
x=
t=
=


2
2  a 
2a
so we got them !!!!
→
v2 = u2 + 2ax
Uniform Accelerated Motion – all together
1 – D Motion with Constant Acceleration
v = u + at
v avg =
u+v
2
x=ut+
x = vavg t
for any motion
a 2
t
2
v2 = u2 + 2ax
In addition to these equations to solve a problem with constant acceleration
you’ll need to introduce your own coordinate system, because displacement,
velocity and acceleration are vectors (they have directions).
From graph velocity vs. time
First 5 seconds of motion:
Slope of the graph =
𝑓𝑖𝑛𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛
𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Conclusion:
slope of the displacement – time graph is velocity
Therefore for the first 5 s:
slope =
Δx
= v avg
Δt
v = 50/5 =10 ms
vavg gives us no details of the motion
between initial and final points.
Average velocity of a particle during the time interval Δt is equal to the slope of the straight
line joining the initial (P) and final (T) position on the position-time graph.
From graph velocity vs. time
First 5 seconds:
Slope of the graph =
𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
Conclusion:
slope of the velocity – time graph is acceleration
Therefore for the first 5 s:
a = 50/5 =10 ms²
Determine its displacement: a is constant, so
𝑥 = 𝑣𝑎𝑣𝑔 𝑡 =
0 + 𝑎𝑡
𝑡 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
2
Conclusion:
Area under a velocity-time graph is the displacement.
During the first 5 s the object has travelled: ½ x 50 x 5 = 125m
𝐴𝐿𝐿 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑎𝑏𝑜𝑢𝑡 𝑔𝑟𝑎𝑝ℎ𝑠
From displacement – time graph we can find velocity
(instantaneous and average) by calculating the slope
The slope is the rate of change of
acceleration (jerk).
Area under the graph v – t is the change
in velocity.
The slope represent acceleration.
The area under graph represent its displacement.
A small quiz
Acceleration can cause: 1. speeding up 2. slowing down
3. and/or changing direction
So beware: both velocity and acceleration are vectors. Therefore
1. if velocity and acceleration (change in velocity) are in the same
direction, speed of the body is increasing.
2. if velocity and acceleration (change in velocity) are in the opposite
directions, speed of the body is decreasing.
3. If a car changes direction even at constant speed it is accelerating.
Why? Because the direction of the car is changing and therefore its
velocity is changing. If its velocity is changing then it must have
acceleration.
This is sometimes difficult for people to grasp when they first
meet the physics definition of acceleration because in everyday
usage acceleration refers to something getting faster.
A stone is rotating around the center of a circle. The
speed is constant, but velocity is not – direction is
changing as the stone travels around, therefore it must
have acceleration.
Velocity is tangential to the circular path at any time.
ACCELERATION IS ASSOCIATED WITH A FORCE!!!
The force (provided by the string) is forcing the stone to move in a
circle giving it acceleration perpendicular to the motion – toward the
CENTER OF THE CIRCLE - along the force. This is the
acceleration that changes velocity by changing it direction only.
When the rope breaks, the stone goes off in the tangential straightline path because no force acts on it.
In the case of moon acceleration is caused by gravitational
force between the earth and the moon. So, acceleration is
always toward the earth. That acceleration is changing
velocity (direction only).
blue arrow –
velocity
red arrow –
acceleration
1. weakening gravitational force would result in the moon
getting further and further away still circling around earth.
2. no gravitational force all of a sudden: there wouldn’t be
acceleration – therefore no changing the velocity (direction) of
the moon, so moon flies away in the direction of the velocity
at that position ( tangentially to the circle).
3. The moon has no speed – it moves toward the earth – accelerated
motion in the straight line - crash
4. High speed – result the same as in the case of weakening gravitational force
Only the right speed and acceleration (gravitational force) would result in
circular motion!!!!!!!
THE PHYSICS
CLASSROOM
Free Fall
Free fall is vertical (up and/or down) motion of a body where
gravitational force is the only or dominant force acting upon it.
(when air resistance can be ignored)
Gravitational force gives all bodies regardless of mass or shape,
when air resistance can be ignored, the same acceleration.
This acceleration is called free fall or gravitational acceleration
(symbol g – due to gravity).
Free fall acceleration at Earth’s surface is about
g = 9.8 m/s2 toward the center of the Earth.
Let’s throw an apple equipped with a speedometer upward
with some initial speed.
That means that apple has velocity u as it leaves our hand.
The speed would decrease by 9.8 m/s every second on the way up,
at the top it would reach zero, and increase by 9.8 m/s for each
successive second on the way down
g depends on how far an object is from the center of the Earth.
The farther the object is, the weaker the attractive gravitational force is,
and therefore the gravitational acceleration is smaller.
At the bottom of the valley you accelerate faster (very slightly) then on
the top of the Himalayas.
Gravitational acceleration at the distance 330 km from the surface
of the Earth (where the space station is) is 7̴ .8 m/s2.
In reality – good vacuum (a container with the air pumped out) can mimic this
situation.
August 2, 1971 experiment was conducted on the Moon – David Scott – he
simultaneously released geologist’s hammer and falcon’s feather. Falcon’s
feather dropped like the hammer. They touched the surface at the same time.
1. Dr. Huff, a very strong lady, throws a ball upward with initial speed of 20 m/s.
How high will it go? How long will it take for the ball to come back?
Givens:
u = 20 m/s
g = - 10 m/s2
at the top v = 0
Unknowns:
t=?
y=?
2. Mr. Rutzen, hovering in a helicopter 200 m above our school suddenly drops his pen.
How much time will the students have to save themselves? What is the velocity/speed of
the pen when it reaches the ground?
Givens:
u = 0 m/s (dropped)
g = 10 m/s2
Unknowns:
t=?
v=?
3. Mrs. Radja descending in a balloon at the speed of 5 m/s above our school
drops her car keys from a height of 100 m.
How much time will the students have to save themselves?
What is the velocity of the keys when they reach the ground?
t=?
v=?
4. Dr. Huff, our very strong lady, goes to the roof and throws a ball upward. The ball leaves her
hand with speed 20 m/s. Ignoring air resistance calculate
a. the time taken by the stone to reach its maximum height
b. the maximum height reached by the ball.
c. the height of the building is 60 m. How long does it take for the ball to reach the ground?
d. what is the speed of the ball as it reaches the ground?
d.
v = u + gt
v = 20 – 10 x 6 = – 40 m/s
speed at the bottom is 40 m/s
Graphs of free fall motion
u = 0 m/s
g = 10 m/s2
Time Velocity Distance
(s)
(m/s)
(m)
v = g t = 10t
y=
+
g 2
t = 5 t2
2
0
0
0
1
10
5
2
20
20
3
4
30
40
45
80
Distance vs. time
40
Distance (m)
velocity (m/s)
Velocity vs. time
30
20
10
0
1
2
3
4
5
Time (s)
constant slope → constant acceleration
80
60
40
20
0
0
1
2
3
Time (s)
4
5
changing slope – changing speed → acceleration
If air resistance can not be neglected, there is
additional force (drag force) acting on the body in
the direction opposite to velocity.
Comparison of free fall with no air resistance and with air resistance
In vacuum
displacement
displacement
In air
velocity
velocity
time
Acceleration is getting
smaller due to air resistance
time and eventually becomes
zero.
When the force of the air
resistance equals gravity, the
object will stop accelerating
and maintain the same
time speed.
It is different for different
bodies.
acceleration
acceleration
time
terminal velocity is maximum
velocity an object can reach
in air/any fluid.
time
time
Air Drag and Terminal Velocity
If a raindrops start in a cloud at a height h = 1200m above the
surface of the earth they hit us at 340mi/h; serious damage would
result if they did. Luckily: there is an air resistance preventing the
raindrops from accelerating beyond certain speed called terminal
speed….
How fast is a raindrop traveling when it hits the ground?
It travels at 7m/s (17 mi/h) after falling approximately only 6 m.
This is a much “kinder and gentler” speed and is far less damaging
than the 340mi/h calculated without drag.
The terminal speed for a skydiver is about
60 m/s (pretty terminal if you hit the deck)