Force Equals Rate of Change of Momentum: Count the Normal Component Only
Impulse questions turn a force over time into a change of momentum, and jets or streams of particles turn a rate of mass flow into a force. The correct momentum change, especially for rebounds, decides almost every answer.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Impulse and the force-time graph
A body of mass 16 kg that feels 50 N for 5 s and then 75 N for 5 s receives 250 + 375 = 625 N s, so its speed from rest after 10 s is 625/16, about 39 m/s. Read the area piece by piece: rectangles and triangles, with areas below the axis counted negative.
Streams: force = (mass per second) x (change in velocity)
A jet of density d, cross-section a and speed v delivers mass a d v each second. If it strikes a wall at angle θ to the normal and rebounds elastically, only the normal component v cosθ reverses, so the force is F = (a d v) x 2 v cosθ = 2 a d v² cosθ. A jet releasing 1 kg/s at 5 m/s that is stopped by a block exerts 5 N, so a 2 kg block accelerates at 2.5 m/s². A disc of mass 1.0 kg floats when bullets of 0.05 kg strike at 10 per second and rebound with the same speed: each bullet changes momentum by 2 m v, so 10 x 2 x 0.05 x v = 1.0 x 9.8 gives v = 9.8 m/s.
Rebound from a wall at an angle
For a ball that hits a wall at an angle to the wall and rebounds with the same speed and angle, the component along the wall is unchanged and the component normal to the wall reverses. A 0.5 kg ball at 48 m/s striking at 30° to the wall has a normal component of 48 sin 30° = 24 m/s, so the momentum change is 2 x 0.5 x 24 = 24 kg m/s and, for a contact time of 0.5 s, the average force is 48 N. Check whether the angle is measured from the wall or from the normal before choosing sine or cosine.
OlympIQ exam lens
- ✓Impulse = ∫F dt = area under the F-t graph = change in momentum
- ✓Jet force = (a d v) x (change of velocity): stopping gives a d v², elastic rebound gives 2 a d v² cosθ at angle θ to the normal
- ✓Only the normal component of velocity reverses in an elastic rebound
- ✓Bullets or particles hitting repeatedly: average force = rate x momentum change per hit
- ✓Check whether the angle is from the wall or the normal
Do not use the full speed in the momentum change when the ball strikes at an angle. Only the component normal to the wall changes; the angle given from the wall means the normal component is v sin of that angle, not v cos.
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