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C. Energy and Momentum · Momentum and Collisions

Momentum and Collisions: Unit Review

C1.1 C1.2 C2.1 C2.5 C2.6 C2.7 C3.3 C3.4 C3.5 A1.1 A1.5 A1.6 A1.8 A1.10 A1.12 A1.13 A2.1 A2.2About 40 minutes
Learning goals
  • I can solve momentum, impulse and collision problems in one and two dimensions.
  • I can decide whether a collision is elastic or inelastic and use energy where it applies.
  • I can explain applications of momentum, from rockets and safety equipment to the neutrino.

Words to know

momentum
p = mv, a vector in the direction of the velocity.
impulse
J = FΔt, equal to the change in momentum.
elastic collision
A collision in which total kinetic energy is conserved.
inelastic collision
A collision in which some kinetic energy is converted to other forms; perfectly inelastic if the objects stick together.

Bringing the unit together

This review covers every lesson of Momentum and Collisions. Each "Try it" set is named after a lesson and mixes quick recall with a calculation. Work each question on paper before you open the answer, then check the parent notes to see which lesson to revisit for each kind of mistake. Use g = 9.8 N/kg, and keep track of directions: momentum and impulse are vectors.

How to use this review

  1. Do one or two sets a day rather than all of them at once.
  2. For every calculation, write the system, the direction you call positive (or your x and y axes), and the law you are using before you substitute numbers.
  3. Mark each question right, close or wrong. Revisit the lesson for any set where you got more than one wrong.
  4. Finish with the 10-question quiz. Aim for 8 or more, then redo the questions you missed a day later.
Just for fun

Momentum is the review topic that never quits: once a system has it, the only way to take it away is with an outside force. If you feel this unit has given you a lot of momentum, conservation says you get to keep it until something external stops you.

momentump = mv (a vector)impulseJ = FΔt = Δpconservationp before = p aftertwo dimensionsadd x and y separatelyforce–time graphsimpulse = areacollisionselastic or inelasticexplosions, rocketstotal p stays zerosafety equipmentlonger Δt, smaller Fneutrinomissing p and energy
How the big ideas of the unit connect: momentum leads to impulse and to conservation, which explain collisions, explosions, rockets and safety equipment. A decay is like an explosion from rest, which is how the neutrino was predicted.

Linear Momentum: Mass in Motion

1

Find the momentum of a 0.16 kg hockey puck moving at 40 m/s [E].

2

A 3000 kg truck moves at 5.0 m/s and a 1200 kg car at 12 m/s. Which has more momentum?

3

A 0.50 kg ball moving at 8.0 m/s [E] bounces back at 6.0 m/s [W]. Find its change in momentum.

Impulse and the Impulse–Momentum Theorem

1

A force of 250 N acts on a ball for 0.040 s. Find the impulse.

2

A 0.145 kg baseball arrives at 38 m/s and leaves the bat at 45 m/s in the opposite direction. Contact lasts 0.70 ms. Find the average force on the ball.

3

Why should you bend your knees when you land from a jump?

Force–Time Graphs: Impulse as an Area

1

A force–time graph is a triangle with a peak of 600 N and a total duration of 0.020 s. Find the impulse.

2

A constant net force of 40 N acts for 0.50 s on a 2.0 kg cart at rest. Find its final speed.

3

Two force–time graphs have the same area, but one is tall and narrow and the other is short and wide. Compare the impulses and the peak forces.

Conservation of Momentum in One Dimension

1

A 2.0 kg cart at 3.0 m/s [E] hits a 1.0 kg cart at rest, and they stick. Find their velocity.

2

A 1500 kg car at 20 m/s [E] collides with a 2500 kg truck at 10 m/s [W], and they lock together. Find their velocity just after.

3

Under what condition is the total momentum of a system conserved?

Explosions and Recoil

1

A 4.0 kg rifle fires a 0.010 kg bullet at 400 m/s. Find the rifle's recoil speed.

2

Two skaters, 60 kg and 40 kg, stand at rest and push apart. The 40 kg skater moves at 3.0 m/s [W]. Find the other's velocity.

3

In an explosion from rest, the total kinetic energy goes from zero to a large value. Where does it come from?

Elastic and Inelastic Collisions

1

A 1.0 kg cart at 4.0 m/s hits a 3.0 kg cart at rest, and they stick. Find the kinetic energy lost and the percentage lost.

2

Before a collision the total kinetic energy is 10.0 J; after it, 10.0 J. Classify the collision.

3

In which type of collision is momentum conserved: elastic, inelastic or both?

Head-On Elastic Collisions

1

A 0.20 kg ball at 3.0 m/s hits a 0.10 kg ball at rest head-on, elastically. Find both final velocities.

2

Two equal masses collide head-on elastically, one of them at rest. What happens?

3

A tennis ball hits a heavy, rigid wall head-on in a nearly elastic collision. Describe its motion after.

Using Momentum and Energy Together: Ballistic Pendulums, Springs and Crash Reconstruction

1

A 0.010 kg bullet embeds in a 2.0 kg pendulum block, which then rises 0.050 m. Find the bullet's speed.

2

A compressed spring between a 1.0 kg cart and a 2.0 kg cart (both at rest) stores 3.0 J. When released, how fast does each cart move?

3

In a crash reconstruction, why is momentum used for the collision itself but energy used for the skid afterward?

Is the net external forceon the system about zero?No: use impulseFΔt = Δp for one objectYes: momentum is conservedwrite p before = p afterIs it 1-D or 2-D?2-D: x and y equationsDo they stick together?perfectly inelastic: one v′Told it is elastic?also conserve EkOtherwise check Ekcompare before and afterBefore or after the collision?use energy (springs, heights, skids)
A decision guide for momentum problems. Start at the top and follow the arrows; it is a guide, so a problem may need more than one path (for example, a ballistic pendulum uses momentum for the collision and energy for the swing).

Momentum in Two Dimensions

1

Add 3.0 kg·m/s [E] and 4.0 kg·m/s [N].

2

Find the components of 10 kg·m/s [E 30° N].

3

A 0.40 kg ball's velocity changes from 5.0 m/s [E] to 5.0 m/s [N]. Find its change in momentum.

Glancing Collisions in Two Dimensions

1

A 1500 kg car at 20 m/s [E] and a 2000 kg truck at 15 m/s [N] collide and lock together. Find their velocity just after.

2

A 0.20 kg puck at 5.0 m/s [E] hits an identical puck at rest. The first leaves at 3.0 m/s [E 53° N]. Find the second puck's velocity.

3

Is the puck collision in the previous question elastic? What is the angle between the pucks' paths afterward?

Investigation: Testing Conservation Laws in Collisions and Explosions

1

In a cart collision, the measured total momentum is 1.20 kg·m/s before and 1.14 kg·m/s after. Find the percentage difference compared with the value before.

2

Suggest one source of error that would make the momentum after a collision consistently smaller than before.

3

Why is a video or a motion sensor better than a hand-held stopwatch for measuring cart speeds?

Rockets and Propulsion: Momentum in Action

1

A rocket engine expels 50 kg of exhaust each second at 2000 m/s relative to the rocket. Find the thrust.

2

Explain why a rocket works in space, where there is no air to push against.

3

A rocket's exhaust speed is 3000 m/s and its initial mass is 3.0 times its final mass. Find its change in speed (ignoring gravity and air resistance).

Momentum and Safety: Vehicles, Helmets and Sports Equipment

1

A helmet's liner increases the stopping time of a head in an impact from 2.0 ms to 8.0 ms. By what factor does it change the average force?

2

Why are cars designed with crumple zones rather than as rigid boxes?

3

Why does a follow-through in a golf swing or hockey slap shot help send the ball or puck faster?

Momentum Technologies, Society and the Environment

1

Give one benefit and one cost of strategic building implosion.

2

Give one environmental concern about fireworks.

3

How does forensic crash reconstruction benefit society?

What Conservation Laws Tell Us: Newton's Cradle, Reactionless Drives and the Neutrino

1

Why did the continuous energy spectrum of beta-decay electrons point to a new particle?

2

A nucleus at rest undergoes beta decay. The electron's momentum is 4.0 × 10⁻²² kg·m/s [E] and the nucleus's is 3.0 × 10⁻²² kg·m/s [S]. Find the neutrino's momentum.

3

In Newton's cradle, three balls are released together at speed v. What happens, and why?

4

Name the Canadian scientist who shared the 2015 Nobel Prize in Physics for work at the Sudbury Neutrino Observatory.

Check your understanding

Quiz

1.

A 2.0 kg ball moves at 3.0 m/s [N]. Its momentum is:

2.

Air bags and crumple zones reduce injuries mainly because they:

3.

The area under a force–time graph equals the:

4.

A 3.0 kg cart at 4.0 m/s hits a 1.0 kg cart at rest, and they stick together. Their speed afterward is:

5.

In a perfectly inelastic collision:

6.

A 50 kg skater at rest throws a 2.0 kg ball forward at 10 m/s. The skater moves backward at:

7.

A moving ball hits an identical ball at rest head-on in an elastic collision. Afterward:

8.

Momentum of 6.0 kg·m/s [E] and 8.0 kg·m/s [N] add to a total of size:

9.

A rocket in deep space speeds up because:

10.

Pauli proposed the neutrino because, in beta decay:

For parents and teachers

How to use the results: have your teen do one or two sets at a time, then the quiz. Use the mistakes to choose what to revisit:

  • Forgetting directions or signs (a bounce gives too small a Δp): Linear Momentum: Mass in Motion.
  • Mixing up force, impulse and Δt (padding, bending knees, follow-through): Impulse and the Impulse–Momentum Theorem and Momentum and Safety: Vehicles, Helmets and Sports Equipment.
  • Trouble finding areas or reading graphs: Force–Time Graphs: Impulse as an Area.
  • Errors setting up p before = p after: Conservation of Momentum in One Dimension (collisions) and Explosions and Recoil (systems starting at rest).
  • Assuming kinetic energy is always conserved, or mixing up elastic and inelastic: Elastic and Inelastic Collisions and Head-On Elastic Collisions.
  • Problems with two stages (a collision then a swing, a spring, or a skid): Using Momentum and Energy Together: Ballistic Pendulums, Springs and Crash Reconstruction.
  • Adding momenta like ordinary numbers in 2-D, or component errors: Momentum in Two Dimensions and Glancing Collisions in Two Dimensions.
  • Percentage difference and error sources: Investigation: Testing Conservation Laws in Collisions and Explosions.
  • Thrust and the rocket equation: Rockets and Propulsion: Momentum in Action.
  • Weak written answers about impacts: Momentum Technologies, Society and the Environment (benefits, costs, alternatives).
  • The neutrino story and Newton's cradle: What Conservation Laws Tell Us: Newton's Cradle, Reactionless Drives and the Neutrino.

Energy topics (work, springs, SHM, damping) are reviewed in Work, Energy and Simple Harmonic Motion.