Download mechanics projectile motion worked example

Physics Summary – Greg Zaal
MECHANICS
PROJECTILE MOTION
When an object is in free fall, the object is at an acceleration of 10m/s down
Displacement is the straight line from start to finish in that direction
Projectile: An object that is launched into the air by throwing, kicking, hitting or firing from a weapon.
Projectile motion: When an object is dropped or thrown up into the air.
An object is experiencing free fall when the only force acting on it is the force of gravity (that means that
there is also no air resistance.
If you throw and object up and catch it:
= ↓= 10.
= 0 . = If you drop something:
=0
↓= 10 . = !"# $ %& ' ! "" (
WORKED EXAMPLE:
A girl standing near the edge of a high cliff throws a stone vertically upwards. The stone leaves her hand at
20 m/s. It falls to the rocks below which are 160m down. Calculate how long it takes for the stone to reach
the bottom of the cliff.
= 20 . *
=∗
= 10 . 1 2(
= 160 1 2(
=?
= + −160 = /200/0 + /−100/ 0
0 = −5 + 20 + 160
0 = − 4 − 32
0 = / − 80/ + 40
∴ = 8 ' = −4
∴ = 8
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Physics Summary – Greg Zaal
PROJECTILE MOTION IN TWO DIMENSIONS
The path of a projectile is called a trajectory.
A trajectory is always parabolic.
To identical objects, one dropped and one flicked, will hit the ground at the same time.
When bodies are in free fall, their horizontal motion does not affect their vertical motion
The vertical motion does not affect the horizontal motion
The vertical and horizontal motions are independent of each other.
HORIZONTAL PROJECTION
The actual velocity can be expressed as a combination of (1 9
Acceleration in the horizontal direction is zero (0)
Acceleration in the vertical direction is the acceleration due to gravity
The horizontal velocity remains constant.
The vertical velocity increases at 10 . every second
The displacement is the resultant of the horizontal displacement and the vertical displacement.
Horizontal displacement is called the Range.
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Physics Summary – Greg Zaal
WORKED EXAMPLE:
John flicks a ball of paper with a horizontal velocity of 2 m/s from the lab bench which is 1.2m high.
Calculate how long it will take for the paper to reach the ground and how far away from the base of the lab
bench the paper will land.
9 = 0 . = + 9 = ∗
9 = 10 . 1 2(
9 = 1:2 9 =?
= 2. = 2. = 0 . = ?
= 0.4;
=<
=>
=
1.2= /00/0 + /100/0
1.2= 5 = 0.24
= 0.4; = + = /20/0.4;0 + /00/0.4; 0
= /20/0.4;0
= 0.;8
?
= < ? 0.4;
= /200.4;
= 0.;8 PROJECTION AT AN ANGLE
When you project an object at an angle, both (1 9 @ 0
A = BCD E
# = DFG E
WORKED EXAMPLE:
A cricketer fielding on the boundary throws the ball with a velocity of 20 m/s at an
angle of 30 degrees to the ground. Calculate how far away the ball will be when it is
caught in order to stump the batsman.
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Physics Summary – Greg Zaal
MOMENTUM
* = Mass in kg
Velocity in . Momentum in HI. . If motion is only in one plane, one direction will be negative while the other positive.
LAW OF CONSERVATION OF MOMENTUM
When bodies in a system interact, their total momentum remains constant in both magnitude and
direction (provided that no external forces act on the system)
+ = + IMPULSE
It is the change in momentum.
Measured in HI. . ' K. Δ* = − WORK
Work is done on an object when a force applied to the object causes it to move in the direction of the force
If you push a car and it moves, work is being done.
If you push a car and it does not move, work is not being done.
If you hit the ball while playing tennis, work is only being done while the racket is in contact with the ball.
Force in N
Displacement in m
Work in K. ' O
2 =L×N
LPQ = LR + LSPT
UR = U/QSR0 + USPT/QRQQ0
UQRQQ = LSPT × N
U = LPQ × N
UR = LR × N
ENERGY
Whenever work is done, energy is transferred or transformed.
Energy is the ability to do work.
It is a scalar quantity.
GRAVITATIONAL
The energy an object has due to its position above the ground.
V = Iℎ
I = acceleration due to gravity = 10. Derivation:
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Physics Summary – Greg Zaal
U = L. N
U = /I0ℎ
U = Iℎ
CHEMICAL
The energy stored in fuels and food stuffs
ELASTIC
The energy stored inside a cable/elastic/rope…
KINETIC
The energy an object has because of its motion.
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VX = 2
LAW OF CONSERVATION OF ENERGY
Energy cannot be created or destroyed, merely changed from one form to another.
MECHANICAL ENERGY
For an object in free fall, projected up, on a swing or a rollercoaster, the mechanical energy is equal to
V + VX and it is conserved.
WORKED EXAMPLE
A 2kg object is dropped from a height of 10m. Ignoring air resistance, calculate the velocity when it is 3m
above the ground.
VT = VT
Iℎ + = Iℎ + /20/100/100 + /20/0
200 + 0 = 60 + = 40
≈ 11:832 . 0
= /20/100/30 + /20/0
WORK-ENERGY THEORY
The net work done on an object is calculated using U = LPQ × N
Fres implies the object is accelerating which means its velocity is changing. Therefore the kinetic energy is
also changing.
U = ∆VX
LPQ . N = − 5
Physics Summary – Greg Zaal
POWER
Power is the rate at which work is done – or – the rate at which energy is transferred or transformed.
U
[=
V
[=
Measured in O. ' U /2
0
FRAMES OF REFERENCE
Relative Velocity is the velocity of an object measured in a specific frame of reference (a set of reference
points that enables the position of an object to be defined at any time)
\]^T PR> P
= \]^T PR> > SP S PSPT + \> SP S PSPT PR> P
WAVES, SOUND AND LIGHT
ELECTROMAGNETIC RADIATION
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The electromagnetic wave is a transverse wave consisting of both an electric field and a magnetic
field component. The two fields oscillate at 90° to each other and the direction of motion.
This wave is formed by charges continuingly changing velocity (accelerating)
A changing magnetic field produces an electric field and a changing electric field produces a
magnetic field.
This wave can travel through a vacuum because it relies on oscillating fields for movement and not
particles.
CHARACTERISTICS OF ELECTROMAGNETIC WAVES
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Travel at the speed of light: 3 × 10_ . (symbol: C)
Transfer energy from one plane to another
When they strike material that absorbs them, they give energy to that material causing the
temperature to increase.
These waves can be reflected, refracted, diffracted and undergo interference
They don’t need a medium to travel through
Consist of two oscillating in phase fields.
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Physics Summary – Greg Zaal
ELECTROMAGNETIC SPECTRUM
Trend
Type of radiation Wavelength (M) Frequency (Hz)
Greatest frequency Gamma
10
3 × 10`
`
X rays
10
3 × 10_
Ultraviolet
10_
3 × 10a
a
Visible Spectrum
10
3 × 10b
b
Infrared
10
3 × 10
Microwaves
10
3 × 10_
Lowest Frequency Radio waves
10=c
3 × 10b
Uses
Electricity Generation
Medical (scans)
Medical (disinfection)
Visibility
TV remotes; heaters
Telecommunication; cooking
Radio communication
The energy of a wave is directly proportional to the frequency of the wave
V = ℎ&
h: Plank’s constant = 6.6 × 10cb
The greater the frequency (and hence energy) of a wave, the grater its ability to penetrate various
materials.
2D AND 3D WAVES
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Mechanical wave: need a medium to travel in. Particles of the wave vibrate at 90° to the movement
of the wave. Eg: air and water (Suncity wavepool – barrier goes up and down in water)
Electromagnetic wave: Do not need a medium. Vibrations are increasing and decreasing intensities
of magnetic and electric fields.
Longitudinal wave: need a medium. Particles vibrate parallel to the direction in which the wave
moves. Creates areas of high pressure (compressions) and low pressure (rarefactions). Eg: sound;
some earthquakes.
Transverse wave: needs a medium. Particles vibrate at 90° to which the wave moves. Eg: slinky
TERMINOLOGY
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Amplitude: distance from equilibrium position to
the crest or the trough
Wavelength: Distance between two consecutive
points in phase
Frequency: Number of waves per second
Period: Time taken for a complete wave to move
past a point
Speed: = &d
Superposition: When two pulses cross, their
combined displacement it equal to the sum of their
individual displacements
Nodal Line: And area where destructive interference has occurred and results in a region of zero
disturbance
Antinodal Line: forms as a result of constructive interference.
Photon: a “packet” of light
Intensity of light: the brightness. The more intense a light, the more photons it has.
Work Function: The minimum amount of energy needed to emit an electron from the surface of a
metal.
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Physics Summary – Greg Zaal
INTERACTION OF ELECTROMAGNETIC RADIATION AND MATTER
THE PHOTOELECTRIC EFFECT
Light of sufficient energy knocks out electrons from the surface of a metal.
Each metal has a threshold frequency – the frequency of the photon of light which is needed to eject an
electron from the surface of a metal.
The greater the frequency of the light, the more energy the ejected electrons will have.
The greater the intensity of the light, the more electrons there will be that are ejected.
V T = U 'HL(!" ( + VH
1
ℎ& = ℎ&PQ + 2
SPECTRUMS
Absorption Spectrum:
1. Electrons in gas absorb photons of light
2. Which are of particular frequencies relative to the gas
3. The energy releases when they fall is released in all directions
4. Hence this energy is unseen (the black areas)
5. As the colour areas are the frequencies which are not absorbed.
Line Emission Spectrum:
1. When electrons gain energy:
2. They become excited and unstable
3. They jump up the energy levels
4. And fall back to a different place
5. Releasing energy
6. And then return to their original position and start again
7. V = ℎ& e hence the different colours
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Physics Summary – Greg Zaal
SOUND
THE DOPPLER EFFECT
It is the apparent change in frequency of a wave when there is relative motion between the source of the
wave and an observer.
When the source is moving towards you, it is a higher pitch.
When it passes you, it is the true pitch only at that moment.
When it moves away, it has a lower pitch.
Source moving towards you: & = <
Source moving away from you:
& = original frequency
& = observed frequency
\ = speed of sound (340 m/s)
= speed of source
f
f>
?&
& =<
f
f=>
?&
WORKED EXAMPLE
An ambulance approaches an intersection at 120 km/h emitting a frequency of 256 Hz. What frequency
will be heard by a stopped driver as the ambulance approaches?
& = 256
& = <
& =?
& = <
\ = 340
= 120 H. ℎ = 33.5. &
f
?&
f>
cb`
cb`cc.c
? /2560
= 283.825 gh
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Physics Summary – Greg Zaal
APPLICATIONS OF THE DOPPLER EFFECT
Medical
• Measure the rate of blood flow (cells reflect waves from transmitted, received by a receiver)
• Used to measure babies heartbeat
Sport
• Measure the speed of a ball – uses a radar gun (radio waves)
Astronomy
• Distance of stars (by the gasses they are made up of and their movement away from us)
SHOCKWAVES
It is a cone shaped wave produced when the
speed of the source is greater than the
speed of the wave.
Sonic boom: The sound heard by an
observer as a shockwave passes.
Eg: Shockwaves in water from a speed boat
Thunder – air expands and compresses fast.
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