Work, Energy & Momentum — Problem set, Grade 12

Account for work, power, stored energy, collisions, and force-time evidence.

Review the key ideas

Work & Power

Work is energy transferred by a force acting through a displacement (W = Fd·cosθ); the net work on an object changes its kinetic energy; power (P = W/t) is how fast that work is done.

Kinetic & Potential Energy

Kinetic energy depends on speed/frame; potential energy depends on interaction and chosen reference. Conserve a consistent energy account rather than mixing force with energy.

Conservation of Energy

Total energy follows a complete boundary account. Mechanical energy remains constant only under appropriate transfer conditions; useful efficiency needs a defined service.

Momentum & Collisions

System momentum change equals external impulse. Negligible external impulse allows momentum conservation; kinetic energy is also conserved only in the elastic collision model.

Impulse & Collision Evidence

Impulse is signed momentum change; the relevant system boundary determines conservation.

Problem 1

A 25 N force pushes a cart 8.0 m in the direction of the force. How much work is done?
Answer and explanation

200 J

W = Fd = 25 × 8.0 = 200 J.

Problem 2

A crane lifts a 700 N crate 3.0 m upward in 5.0 s. Find the work done and the power output.
Answer and explanation

Work = 2,100 J; power = 420 W

Work against gravity: W = Fd = 700 × 3.0 = 2,100 J. Power: P = W/t = 2,100 ÷ 5.0 = 420 W.

Problem 3

A car cruises at a constant 30 m/s while its engine supplies 12,000 W of power to overcome resistance. Using P = Fv, find the total resistive force.
Answer and explanation

400 N

P = Fv, so F = P/v = 12,000 ÷ 30 = 400 N. (At constant speed the driving force equals the resistive force.)

Problem 4

Find the kinetic energy of a 5.0 kg object moving at 4.0 m/s.
Answer and explanation

40 J

KE = ½mv² = ½ × 5.0 × 4.0² = ½ × 5.0 × 16 = 40 J.

Problem 5

A spring with spring constant k = 300 N/m is compressed 0.10 m. How much elastic potential energy is stored?
Answer and explanation

1.5 J

PE = ½kx² = ½ × 300 × 0.10² = ½ × 300 × 0.01 = 1.5 J.

Problem 6

A 500 kg roller-coaster car starts from rest at the top of a 40 m hill. Ignoring friction, find (a) its gravitational potential energy at the top and (b) its speed at the bottom. (g = 9.8 m/s²)
Answer and explanation

PE = 196,000 J; speed = 28 m/s

PE = mgh = 500 × 9.8 × 40 = 196,000 J. All of it becomes kinetic energy: 196,000 = ½ × 500 × v² = 250v², so v² = 784 and v = 28 m/s.

Problem 7

A device is supplied 1,200 J of energy and produces 900 J of useful output. Find its efficiency and the energy lost as waste heat.
Answer and explanation

75% efficient; 300 J lost

Efficiency = (900 ÷ 1,200) × 100% = 75%. Energy lost = 1,200 − 900 = 300 J, dissipated as heat.

Problem 8

The net force on a 5.0 kg cart, initially at rest, does 250 J of work on it. Using the work-energy theorem, find its final speed.
Answer and explanation

10 m/s

W_net = ΔKE, so 250 = ½ × 5.0 × v² = 2.5v². Then v² = 100 and v = 10 m/s.

Problem 9

A pump lifts 200 kg of water to a tank 15 m higher, using 40,000 J of electrical energy. Find (a) the useful work done on the water and (b) the pump's efficiency. (g = 9.8 m/s²)
Answer and explanation

Useful work = 29,400 J; efficiency = 73.5%

Useful work = gain in gravitational PE = mgh = 200 × 9.8 × 15 = 29,400 J. Efficiency = (29,400 ÷ 40,000) × 100% = 73.5%.

Problem 10

Find the momentum of a 2.0 kg ball moving at 8.0 m/s.
Answer and explanation

16 kg·m/s

p = mv = 2.0 × 8.0 = 16 kg·m/s.

Problem 11

A 1.0 kg ball moving at 6.0 m/s strikes an identical 1.0 kg ball at rest in a head-on elastic collision. Find the velocity of each ball afterward.
Answer and explanation

The incoming ball stops (0 m/s); the struck ball moves off at 6.0 m/s.

For a head-on elastic collision between equal masses, the objects exchange velocities. Check: momentum 1.0 × 6.0 = 1.0 × 6.0 ✓, and kinetic energy ½ × 1.0 × 6.0² = ½ × 1.0 × 6.0² ✓. This is exactly the behavior seen in a Newton's cradle.

Problem 12

A 2.0 kg cart moving at 5.0 m/s catches up to a 3.0 kg cart moving at 1.0 m/s in the same direction and couples to it. Find their common velocity.
Answer and explanation

2.6 m/s

Conservation of momentum: 2.0 × 5.0 + 3.0 × 1.0 = (2.0 + 3.0)v′, so 10 + 3 = 5.0v′, giving 13 = 5.0v′ and v′ = 2.6 m/s.

Problem 13

A 0.20 kg ball hits a wall at 5.0 m/s and bounces straight back at 5.0 m/s. Find the magnitude of the impulse the wall gives the ball.
Answer and explanation

2.0 kg·m/s

Taking the outgoing direction as positive, Δp = m(v_f − v_i) = 0.20 × (5.0 − (−5.0)) = 0.20 × 10 = 2.0 kg·m/s. (Bouncing back doubles the impulse compared with simply stopping.)

Problem 14

A10N parallel force moves a box 3 m in 2 s. Find work and average power.
Answer and explanation

W=30 J; P=W/t=15W, if that force is constant.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 15

Chapter review: A motor raises a 20 kg load by 3 m in 6 s at 70% efficiency. Find useful work, output power and input energy.
Answer and explanation

Useful work=mgh=588 J. Output power=98 W. Input energy=588/0.70=840 J; other transfers/storage total252 J under the stated boundary.

Useful work=mgh=588 J.

Output power=98 W.

Input energy=588/0.70=840 J; other transfers/storage total252 J under the stated boundary.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 16

A2 kg mass at 3 m has what U usingg=9.8 andU=0 atground?
Answer and explanation

U=mgh=58.8 J; changing the reference shifts absolute U but not energy differences.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 17

Chapter review: A1.5 kg cart has speed4 m/s. It compresses a loss-free spring with k=600 N/m on a level track. Find initial K and maximum compression.
Answer and explanation

K=½×1.5×16=12 J. Set12=½×600x², giving x=0.20 m within the spring’s assumed elastic range.

K=½×1.5×16=12 J.

Set12=½×600x², giving x=0.20 m within the spring’s assumed elastic range.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 18

A cart loses 20 J mechanical energy due to friction. Where is it?
Answer and explanation

Energy transfers into thermal energy, deformation, or sound of the chosen larger system; account for boundaries.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 19

Chapter review: A system begins with200 J mechanical energy. It ends with120 J mechanical,50 J thermal increase and20 J transferred outward as sound/heat. Find the unaccounted energy and a useful next check.
Answer and explanation

The accounted final/output total is190 J, leaving10 J. Check boundary heat transfer, deformation/storage and instrument uncertainty rather than claiming energy was destroyed.

The accounted final/output total is190 J, leaving10 J.

Check boundary heat transfer, deformation/storage and instrument uncertainty rather than claiming energy was destroyed.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 20

2 kg at 3 m/s sticks to 1 kg atrest. Find final velocity.
Answer and explanation

Initialp=6 kg·m/s, totalm=3 kg, so vf=2 m/s with negligible external impulse.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 21

Chapter review: A3 kg cart at4 m/s sticks to a1 kg cart initially at rest. Neglect external impulse. Find final velocity and kinetic-energy decrease.
Answer and explanation

Initial momentum=12 kg·m/s, so vf=3 m/s. Ki=24 J and Kf=18 J;6 J leaves the kinetic category through other energy forms.

Initial momentum=12 kg·m/s, so vf=3 m/s.

Ki=24 J and Kf=18 J;6 J leaves the kinetic category through other energy forms.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

Problem 22

A triangular force pulse has base .020 s and height 100N. Find impulse.
Hints

Write the relationship first; substitute with consistent units.

Answer and explanation

1 N·s

.5×.020×100=1.

Problem 23

Why define the system for collision conservation?
Answer and explanation

External forces can transfer momentum into or out of the selected system.

Response criteria

  • State a scientifically accurate conclusion.
  • Show the calculation or mechanism that supports it.
  • Include units, conditions, or limits where relevant.

Problem 24

Chapter review: A0.50 kg object slows from6 m/s to0 over0.15 s. Find impulse and average net force. If it rebounds at−2 m/s instead, find the new impulse.
Answer and explanation

Stopping impulse=0.50(0−6)=−3 N·s and Favg=−20 N. With rebound, impulse=0.50(−2−6)=−4 N·s, so the fixed-impulse comparison no longer applies.

Stopping impulse=0.50(0−6)=−3 N·s and Favg=−20 N.

With rebound, impulse=0.50(−2−6)=−4 N·s, so the fixed-impulse comparison no longer applies.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical or environmental mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.
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