Momentum

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レッスンノート

Momentum Basics

Momentum is a property of moving objects, defined by the equation p = m v.
p = momentum in kilogram metre per second (kg m/s), m = mass in kilograms (kg), v = velocity in metres per second (m/s).
An object at rest has v = 0, so its momentum is zero.
Momentum is a vector quantity: it has both size and direction, so it can be positive or negative.
If an object moves to the right, its momentum is positive; if it moves to the left, its momentum is negative.
Momentum changes if an object speeds up, slows down, changes direction, or its mass changes.
The units of momentum are kg m/s, which come from multiplying mass (kg) by velocity (m/s).

Conservation of Momentum

The law of conservation of momentum states: in a closed system, the total momentum before an event equals the total momentum after the event.
A closed system means no external forces act (e.g. friction is absent) and the total energy remains constant.
For a collision: total momentum before = total momentum after.
Momentum is always conserved over time, in every collision or explosion.
Because momentum is a vector, objects moving in opposite directions at the same speed have an overall momentum of zero (they cancel out).
In calculations, choose a positive direction (usually right or upwards) and give opposite directions a negative sign.

Conservation of momentum

Collisions

In a collision, objects either bounce apart and move in opposite directions, or stick together and move in the same direction.
When objects move in opposite directions, each has a different velocity depending on its mass and the initial momentum of the system.
When objects stick together, they have a combined mass and a combined velocity after the collision.
An elastic collision is one where the objects move in opposite directions; an inelastic collision is one where the objects stick together and move in the same direction.
In a perfectly elastic collision, kinetic energy is the same before and after; in an inelastic collision, kinetic energy is not conserved (it may become heat, sound, etc.).
To solve collision problems, use conservation of momentum: set total momentum before equal to total momentum after and solve for the unknown velocity or mass.

A car moving towards a stationary van before a collision.

Force and Momentum

When a force acts on a moving object (or one able to move), it accelerates or decelerates, causing a change in momentum.
Force is the rate of change of momentum: F = Δp / Δt, where Δp is the change in momentum and Δt is the time taken.
The change in momentum is final momentum minus initial momentum: Δp = mv – mu (where u is initial velocity and v is final velocity).
This equation comes from combining F = m a and a = (v – u) / t.
Force and momentum are vectors, so they can be positive or negative depending on direction.
For a given change in momentum, a longer contact time means a smaller force; a shorter contact time means a larger force.

A person holding an umbrella in rain versus hail, illustrating the difference in force and momentum change.

Momentum and Safety

Safety features increase the contact time during an impact, which reduces the force on the person because force is the rate of change of momentum.
Seat belts stretch slightly to increase the time for the passenger's momentum to reach zero, reducing the force on them.
Airbags deploy on impact and act as a soft cushion, increasing the time over which the passenger's momentum changes.
Crumple zones at the front and back of vehicles are designed to crush in a controlled way, increasing the time for the vehicle to come to rest and lowering the impact force.
Crash mats in gymnasiums and bouldering mats are thick and soft, increasing contact time when a person lands, reducing the impact force.
Cushioned playground surfaces increase contact time when a child falls, reducing the risk of severe injury.
The effectiveness of safety equipment depends on mass and velocity: a large mass moving fast has a large momentum, so a very long contact time is needed to reduce the force.

Explosions and Recoil

In an explosion starting from rest, the total momentum before is zero, so the total momentum after must also be zero.
This means the objects move apart in opposite directions with equal and opposite momenta.
For example, when a gun fires a bullet, the bullet moves forward and the gun recoils backwards.
The momentum of the bullet forward equals the momentum of the gun backwards in size, but opposite in direction.
This principle also explains why a person jumping off a trolley causes the trolley to move in the opposite direction.

Newton's Third Law and Momentum

Newton's third law states: whenever two bodies interact, the forces they exert on each other are equal and opposite.
In a collision, when object A exerts a force on object B, object B exerts an equal force on object A in the opposite direction.
These forces cause one object to gain momentum and the other to lose momentum.
For objects of equal mass, the accelerations are equal; for unequal masses, the accelerations are unequal (the lighter object accelerates more).
The two forces in Newton's third law always act on different objects.

Key Skills and Common Mistakes

Always draw a before-and-after diagram to keep track of masses, velocities, and directions.
Choose a positive direction and stick to it; give opposite velocities negative signs.
Remember that momentum is a vector – do not treat it as a scalar.
Momentum is conserved in all collisions and explosions, not just elastic ones.
Do not confuse momentum with kinetic energy or force – they are different quantities.
When calculating force from momentum change, use F = Δp / Δt and ensure Δt is the contact time.
Double-check signs in your final answer – sign errors are the most common mistake.

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練習問題

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1.Which equation correctly relates momentum (p), mass (m) and velocity (v)?
Easy
- Ap = mv
- Bp = m/v
- Cp = v/m
- Dp = m + v
2.What are the correct units for momentum?
Easy
- Akg m/s
- Bkg/m/s
- CN m
- Dkg m/s²
3.A tennis ball and a brick are compared. The brick is much heavier, but the ball travels much faster. If they have the same momentum, what can be said about the force they exert on impact?
Medium
- AThey would exert a similar force if they take the same time to come to rest
- BThe brick always exerts a greater force because it is heavier
- CThe ball always exerts a greater force because it is faster
- DNeither exerts any force because momentum is the same
4.Which statement best describes the principle of conservation of momentum?
Medium
- AIn a closed system, total momentum before an event equals total momentum after the event
- BMomentum is only conserved when objects bounce off each other
- CMomentum is always conserved, even when external forces act on the system
- DTotal momentum before a collision is always greater than after
5.Which of the following are true about momentum? (Select all that apply)
Medium
- AMomentum is a vector quantity
- BMomentum can be negative if the object moves in the opposite direction to the positive direction
- CAn object at rest has zero momentum
- DMomentum is the same as kinetic energy
- EMomentum is measured in newtons
6.Momentum is a scalar quantity.
Easy
True or false?
7.Match each term with its correct definition.
Medium
- Momentum
- Conservation of momentum
- Closed system
- Impulse
- The product of mass and velocity
- Total momentum before an event equals total momentum after
- A system with no external forces and constant energy
- Force multiplied by the time for which it acts
8.A car of mass 860 kg is travelling at 30 m/s. Calculate its momentum.
Medium
- A25 800 kg m/s
- B28.7 kg m/s
- C860 kg m/s
- D25 800 N

もっと問題

9.Which of the following equations relating momentum (p), mass (m) and velocity (v) are correct? (Select all that apply)
Medium
- Ap = mv
- Bv = p/m
- Cm = p/v
- Dp = m/v
- Ev = m/p
10.A paintball gun of mass 0.5 kg fires a paintball forwards. The paintball has a mass of 2.5 g and leaves the gun at 70 m/s. What is the momentum of the paintball just after the gun is fired?
Medium
- A0.175 kg m/s
- B175 kg m/s
- C0.000175 kg m/s
- D1.75 kg m/s
11.A boulderer falls for 1.01 s before hitting a crash pad. Acceleration due to gravity is 9.81 m/s². Calculate his speed before hitting the crash pad. Give your answer to 3 significant figures.
Medium
- A9.91 m/s
- B9.81 m/s
- C10.0 m/s
- D0.103 m/s
12.Car A has a mass of 600 kg and travels at 30 miles/hour to the right. The total momentum of the system of car A and car B is 58 000 kg miles/hour. Car B travels at 40 miles/hour to the right. Calculate the mass of car B. Give your answer to 2 significant figures.
Medium
- A1000 kg
- B1450 kg
- C600 kg
- D550 kg
13.A girl of mass 36 kg jumps backwards off a stationary trolley of mass 7.5 kg. The trolley moves backwards at 1.2 m/s. Calculate the velocity at which the girl moves forwards.
Medium
- A0.25 m/s
- B0.10 m/s
- C2.5 m/s
- D0.50 m/s
14.Which of the following are true about conservation of momentum? (Select all that apply)
Medium
- AIt applies in a closed system where no external forces act
- BThe total momentum before an event equals the total momentum after
- CIt applies to explosions as well as collisions
- DIt only applies to elastic collisions
- EMomentum is lost during an inelastic collision
15.In an explosion that starts from rest, the total momentum of the system after the explosion is zero.
Easy
True or false?
16.Momentum is only conserved during elastic collisions.
Medium
True or false?
17.Match each safety feature with how it reduces injury.
Medium
- Seat belt
- Airbag
- Crumple zone
- Crash mat
- Stops the passenger colliding with the interior and stretches to increase the time for their momentum to reach zero
- Acts as a soft cushion to prevent injury when the passenger is thrown forward
- Crushes or crumples in a controlled way to increase the time the vehicle takes to come to rest
- Is thick and soft to absorb the force of a landing over a longer contact time
18.Place these steps in the correct order to calculate the velocity of a van pushed forward by a car that collides with it.
Medium
- State the principle of conservation of momentum
- Calculate the total momentum before the collision
- Write the total momentum after the collision in terms of the unknown velocity
- Rearrange and solve for the velocity of the van
19.A moving object's momentum will change if... (Select all that apply)
Medium
- AThe object accelerates or decelerates
- BThe object changes direction
- CThe object's mass changes
- DThe object stays at rest
- EThe object moves at a constant velocity in a straight line
20.A tennis ball hits a racket with a change in momentum of 0.5 kg m/s. Racket A is in contact with the ball for 0.02 s and racket B for 0.05 s. Which racket experiences the greater force from the ball?
Hard
- ARacket A, because the contact time is shorter
- BRacket B, because the contact time is longer
- CBoth experience the same force
- DRacket B, because the change in momentum is larger
21.What is the value of the initial momentum of the system before the explosion?
Medium
- A0 kg m/s
- B80 kg m/s
- C−240 kg m/s
- DIt cannot be determined
22.Which statement describes what is meant by the conservation of momentum in this collision?
Medium
- AThe total momentum of the two trolleys before the collision equals the total momentum after
- BThe momentum of each trolley stays the same
- CThe kinetic energy of the trolleys is conserved
- DThe forces between the trolleys are equal and opposite
23.Which of the following are types of event in which conservation of momentum applies? (Select all that apply)
Medium
- AA collision between two vehicles
- BAn explosion
- CA rocket launching
- DA ball rolling at constant speed with no other objects
- EA stationary object staying still
24.A 1200 kg car travelling at 20 m/s collides with a stationary 800 kg car. The two cars stick together and move off together. What is their combined velocity?
Hard
- A12 m/s
- B10 m/s
- C20 m/s
- D30 m/s
25.A tennis ball hits a racket with a change in momentum of 0.5 kg m/s. Racket A is in contact with the ball for 0.02 s and racket B for 0.05 s. Which racket experiences the greater force?
Medium
- ARacket A
- BRacket B
- CBoth experience the same force
- DCannot tell without knowing the mass of the ball
26.A 0.5 kg paintball gun fires a 2.5 g paintball forwards at 70 m/s. What is the momentum of the paintball just after firing?
Medium
- A0.175 kg m/s
- B175 kg m/s
- C0.175 g m/s
- D35 kg m/s
27.A 1.5 kg skateboard moves at 8.5 m/s. What is its momentum to 2 significant figures?
Medium
- A13 kg m/s
- B12.75 kg m/s
- C5.7 kg m/s
- D0.18 kg m/s
28.A 860 kg car slows from 30 m/s to 10 m/s in 2.5 s. What is the braking force?
Medium
- A6880 N
- B3440 N
- C10320 N
- D688 N
29.Which of the following statements about a closed system are correct? (Select all that apply)
Hard
- AThe total momentum before an event equals the total momentum after the event.
- BA closed system has no external forces acting on it.
- CMomentum is only conserved in elastic collisions.
- DThe total energy within a closed system is constant.
- EMomentum is a scalar quantity in a closed system.
30.A 2 kg object moving at 3 m/s collides with a stationary 4 kg object. After the collision, the 2 kg object moves at 1 m/s in the same direction. What is the velocity of the 4 kg object?
Hard
- A1 m/s in the same direction
- B2 m/s in the same direction
- C0.5 m/s in the same direction
- D1 m/s in the opposite direction
31.A 600 kg car travelling at 30 miles/hour collides with a 1000 kg car travelling at 40 miles/hour in the same direction. They stick together. What is their combined speed?
Hard
- A36 miles/hour
- B35 miles/hour
- C34 miles/hour
- D70 miles/hour
32.In a laboratory experiment, two trolleys collide on a horizontal air track. Which piece of apparatus is used to measure the velocity of a trolley as it passes?
Easy
- ALight gate
- BNewton meter
- CThermometer
- DAmmeter
33.A student uses a light gate to measure the velocity of a trolley of length 0.20 m. The gate records a time of 0.050 s. What is the velocity of the trolley?
Medium
- A4.0 m/s
- B0.010 m/s
- C10 m/s
- D0.25 m/s
34.A student investigates a collision between two trolleys on an air track. Which of the following should be done to obtain valid results? (Select all that apply)
Medium
- AUse light gates or data loggers to measure velocities accurately
- BEnsure the air track is level so no external force acts
- CRepeat the experiment and take an average
- DPush one trolley harder to increase its mass
- EIgnore the direction of motion of each trolley
35.In a collision between two objects, the total momentum is always conserved provided no external forces act.
Medium
True or false?
36.Match each practical step in a trolley collision experiment with its purpose.
Medium
- Level the air track
- Use light gates
- Repeat the collision several times
- Measure the mass of each trolley
- To remove the effect of an external force
- To measure velocity accurately
- To reduce the effect of random errors
- To allow momentum to be calculated
37.Place the steps in the correct order to find the velocity of a trolley after a collision using conservation of momentum.
Medium
- Write down the principle of conservation of momentum
- Calculate the total momentum before the collision
- Write an expression for the total momentum after the collision
- Set the momentum before equal to the momentum after
- Rearrange and solve for the unknown velocity
38.A 1200 kg car travelling at 20 m/s collides with a stationary 800 kg car. They stick together and move off together. What is their combined velocity?
Hard
- A12 m/s
- B20 m/s
- C8 m/s
- D10 m/s
39.A car of mass 1200 kg travelling at 15 m/s collides with a stationary car of mass 800 kg and they stick together. Which statement about the kinetic energy of the system is correct?
Hard
- AKinetic energy decreases because the collision is inelastic
- BKinetic energy increases because the cars stick together
- CKinetic energy stays the same because momentum is conserved
- DKinetic energy becomes zero because the cars stop
40.Which of the following is the correct definition of momentum?
Easy
- AThe product of mass and velocity
- BThe rate of change of velocity
- CThe product of force and time
- DThe ratio of force to acceleration
41.What is the name of the principle that states total momentum before an event equals total momentum after, in a closed system?
Easy
42.Complete the sentence about elastic and inelastic collisions.
Easy
When a collision is ____, the objects move in opposite directions. The velocities of the objects are ____. When a collision is ____, the objects move in the same direction together. The velocities of the objects are ____.
43.A skateboard of mass 1.5 kg moves at 8.5 m/s. Calculate its momentum. Give your answer to 2 significant figures.
Medium
44.An oil barrel of mass 80 kg moves to the left with a velocity of −3.0 m/s. Calculate the momentum of the oil barrel. Give your answer to 2 significant figures.
Medium
45.A car and driver have a combined mass of 860 kg. The car slows from 30 m/s to 10 m/s in the positive direction. Calculate the change in momentum of the car. Forward is positive. Give your answer to 2 significant figures.
Hard
46.A car of combined mass 860 kg slows from 30 m/s to 10 m/s in 2.5 s. Calculate the braking force applied. Give your answer to 2 significant figures.
Hard
47.A boulderer of mass 75.5 kg hits a crash pad at 9.91 m/s and comes to rest in 0.00925 s. Calculate the force the crash pad exerts on the boulderer. Give your answer to 3 significant figures.
Hard
48.Car A (600 kg) travels at 30 miles/hour and car B (1000 kg) travels at 40 miles/hour in the same direction. They collide and stick together. Calculate their combined speed. Give your answer to 2 significant figures.
Hard
49.What name is given to the property of an object that describes how difficult it is to change its velocity?
Medium
50.A car of mass 1200 kg is travelling at 15 m/s. It collides with a stationary car of mass 800 kg and they stick together. Calculate their combined velocity after the collision.
Medium
51.Complete the sentences about collisions. Choose answers from: elastic, inelastic, identical, different.
Medium
When a collision is ____ the objects move in opposite directions. The velocities of the objects are ____. When a collision is ____ the objects move in the same direction together. The velocities of the objects are ____.
52.The oil barrel has a mass of 80 kg and moves to the left with a velocity of −3.0 m/s. Calculate the momentum of the oil barrel.
Medium
53.Explain why momentum is a vector quantity.
Medium
54.Show that kinetic energy is not conserved in the collision where car A (600 kg, 30 miles/hour) and car B (1000 kg, 40 miles/hour) stick together and move off at 36 miles/hour.
Hard
55.While landing, a boulderer bends his knees. Explain how this reduces the chance of injury.
Medium
56.A car driver who was not wearing a seatbelt is injured when his vehicle collides with a stationary vehicle. Explain why he was injured and how a seatbelt could have reduced the risk of injury.
Hard
57.A car of mass 1200 kg travelling at 20 m/s collides with a stationary car of mass 800 kg. The two cars stick together. Explain, using momentum, why the combined velocity is less than 20 m/s.
Hard
58.A skateboard of mass 1.5 kg moves at 8.5 m/s. Calculate its momentum. Give your answer to 2 significant figures.
Medium
- A14 kg m/s
- B13 kg m/s
- C12 kg m/s
- D15 kg m/s
59.An oil barrel of mass 80 kg moves to the left with a velocity of −3.0 m/s. Calculate the momentum of the oil barrel. Give your answer to 2 significant figures.
Medium
- A-216 kg m/s
- B-240 kg m/s
- C-264 kg m/s
- D-192 kg m/s
60.A car and driver have a combined mass of 860 kg. The car slows from 30 m/s to 10 m/s in the positive direction. Calculate the change in momentum of the car. Forward is positive. Give your answer to 2 significant figures.
Hard
- A-17000 kg m/s
- B-13600 kg m/s
- C-18700 kg m/s
- D-15300 kg m/s
61.A car of combined mass 860 kg slows from 30 m/s to 10 m/s in 2.5 s. Calculate the braking force applied. Give your answer to 2 significant figures.
Hard
- A-5520 N
- B-6210 N
- C-7590 N
- D-6900 N
62.A boulderer of mass 75.5 kg hits a crash pad at 9.91 m/s and comes to rest in 0.00925 s. Calculate the force the crash pad exerts on the boulderer. Give your answer to 3 significant figures.
Hard
- A-64720 N
- B-72810 N
- C-80900 N
- D-88990 N
63.Car A (600 kg) travels at 30 miles/hour and car B (1000 kg) travels at 40 miles/hour in the same direction. They collide and stick together. Calculate their combined speed. Give your answer to 2 significant figures.
Hard
- A40 miles/hour
- B36 miles/hour
- C32 miles/hour
- D44 miles/hour
64.A car of mass 1200 kg is travelling at 15 m/s. It collides with a stationary car of mass 800 kg and they stick together. Calculate their combined velocity after the collision.
Medium
- A10 m/s
- B11 m/s
- C8 m/s
- D9 m/s
65.The oil barrel has a mass of 80 kg and moves to the left with a velocity of −3.0 m/s. Calculate the momentum of the oil barrel.
Medium
- A-240 kg m/s
- B-216 kg m/s
- C-192 kg m/s
- D-264 kg m/s

フラッシュカード

State the equation that defines momentum.

p = mv, where p is momentum, m is mass and v is velocity.

What are the units of momentum?

Kilogram metre per second (kg m/s).

What is the momentum of an object at rest?

Zero, because v = 0 so p = mv = 0.

Why is momentum a vector quantity?

Because velocity is a vector, so momentum depends on direction as well as size.

If an object moving to the right has positive momentum, what is the momentum of an object moving to the left?

Negative.

Give three changes that will cause the momentum of an object to change.

It accelerates or decelerates, it changes direction, or its mass changes.

State the principle of conservation of momentum.

In a closed system, the total momentum before an event is equal to the total momentum after the event.

What is meant by a closed system?

A system in which the energy is constant and there are no external forces acting, such as friction.

In a collision, what is always true about momentum?

Momentum is always conserved — total momentum before equals total momentum after.

Describe what happens to two objects in an elastic collision.

They collide and move apart in opposite directions, each with its own velocity depending on its mass and the initial momentum of the system.

Describe what happens to two objects in an inelastic collision.

They stick together and move off in the same direction with a combined mass and a shared velocity.

Two objects of equal mass move towards each other at the same speed. What is the total momentum of the system?

Zero, because their momenta are equal in size but opposite in direction and cancel out.

Write the equation linking force, change in momentum and time.

F = Δp / Δt, where F is force, Δp is the change in momentum and Δt is the time taken.

How is the change in momentum defined?

Final momentum minus initial momentum (Δp = mv − mu).

In words, what does F = Δp / Δt tell you about force?

Force is the rate of change of momentum.

Explain why increasing the contact time of an impact reduces the force on a person.

For a given change in momentum, force is inversely proportional to the time taken, so a longer contact time gives a smaller impact force.

How does a seat belt reduce the risk of injury in a crash?

It stops the passenger hitting the interior of the vehicle and stretches slightly, increasing the time for their momentum to reach zero and so reducing the force on them.

How does an airbag reduce the risk of injury?

It acts as a soft cushion that increases the contact time as the passenger's momentum falls to zero, reducing the force on them.

How does a crumple zone reduce the force on passengers?

It crushes or crumples in a controlled way, increasing the time over which the vehicle comes to rest and so lowering the impact force.

Why do crash mats reduce the chance of injury in a fall?

They are thick and soft, so the body is in contact with the mat for longer, increasing the contact time and creating a smaller impact force.

Why can a child use a thinner crash mat than an adult?

A child has a lower mass, so the change in momentum in a fall is smaller and a shorter contact time is enough to keep the force low.

A skateboard of mass 1.5 kg moves at 8.5 m/s. Calculate its momentum to 2 significant figures.

p = mv = 1.5 × 8.5 = 12.75 ≈ 13 kg m/s.

A car and driver of combined mass 860 kg travel at 30 m/s. Calculate the initial momentum to 2 significant figures.

p = mv = 860 × 30 = 25 800 ≈ 26 000 kg m/s.

A paintball of mass 2.5 g is fired forwards at 70 m/s. Calculate its momentum.

p = mv = 0.0025 × 70 = 0.175 kg m/s.

A gun and paintball are at rest before firing. What is the total momentum of the system after firing?

Zero, because momentum is conserved and the total momentum before firing was zero.

A girl jumps backwards off a stationary trolley. Explain what happens to the trolley.

The total momentum must stay at zero, so the trolley moves forwards with equal and opposite momentum to the girl.

A 36 kg girl jumps off a 7.5 kg trolley that moves backwards at 1.2 m/s. Calculate the girl's forward velocity.

Momentum of trolley = 7.5 × 1.2 = 9 kg m/s, so the girl's momentum is 9 kg m/s and v = 9 ÷ 36 = 0.25 m/s.

A 1 kg object moving at 3 m/s hits a stationary 2 kg object and they stick together. Calculate their combined velocity.

Momentum before = 1 × 3 = 3 kg m/s; combined mass = 3 kg; v = 3 ÷ 3 = 1 m/s.

When two objects collide and stick together, how do you find the momentum after the collision?

Use the combined mass multiplied by their shared velocity.

What is the most common error to avoid in conservation of momentum calculations?

Getting the signs wrong for velocities — objects moving in opposite directions must have opposite signs.

How is kinetic energy different from momentum in a collision?

Momentum is always conserved in a collision, but kinetic energy is not always conserved — it may be transferred to heat, sound or elastic potential energy.

In a perfectly elastic collision, what happens to the kinetic energy?

It is the same before and after the collision.

In a perfectly inelastic collision, what happens to the objects?

They stick together after colliding.

State Newton's third law of motion.

Whenever two bodies interact, the forces they exert on each other are equal in size and opposite in direction.

When two trolleys collide, how do the forces on them compare?

The forces are equal in magnitude and opposite in direction, so F(A on B) = −F(B on A).

Two colliding objects have unequal masses. How do their accelerations compare?

They are unequal, because acceleration depends on both force and mass; the less massive object accelerates more.

Define inertia.

The tendency of an object to continue in its state of rest, or in uniform motion, unless acted upon by an external force.

Define inertial mass.

The ratio of the force applied to an object to the acceleration it experiences (m = F / a).

Three objects are pushed with the same force and accelerate at 1.5, 0.7 and 2.0 m/s². Which has the largest inertial mass?

The one accelerating at 0.7 m/s², because inertial mass is inversely proportional to acceleration for a given force.

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