Impulse and a change of momentum

Impulse measures the effect of a force over an interval. Its direction matters: choose the body and positive direction before finding the area or the momentum change.

Momentum is the vector p = mv. In one dimension, use signed velocities along one axis. Newton's second law relates the resultant force component to the rate of change of that momentum component.

For constant mass, Δp = m(v - u). Impulse has units N s = kg m/s. It is force multiplied by time, not force divided by time.

Check what force the graph represents. A contact-force pulse alone gives the net impulse in that direction only when other impulses in that direction are negligible. For a graph that crosses zero, add its signed positive and negative areas; do not add all area magnitudes.

Estimate an impact force

Assume a roughly 0.1 kg ball is stopped from about 10 m/s. Its momentum change has magnitude of order 1 N s. If contact lasts about 0.01 s, the average force has magnitude of order 100 N, opposite the incoming direction.

These rough mass, speed and duration assumptions give a size check, not a universal impact force. A different contact duration changes the average force even if the momentum change is the same.

Average force is not peak force

The average force gives the same impulse when multiplied by the full interval. Its magnitude, 80.0 N, differs from the pulse's 160 N peak.

If the same -4.80 N s momentum change occurred in 0.030 s, the average force would be -160 N. Increasing contact time can reduce the magnitude of average force for a fixed momentum change. It does not, by itself, specify the peak or the exact pulse shape.

If the original ball merely stopped, its momentum change would be 0 - 3.00 = -3.00 N s. Rebounding adds momentum in the opposite direction, increasing the change's magnitude to 4.80 N s.

The ball also exerts an opposite force on the wall. These Newton's-third-law forces act on different bodies. They do not cancel in the ball's individual momentum equation.

Compare a sensor trace with momentum change

To compare a recorded contact pulse with m(v - u), identify the force recipient and use the same positive direction for both measurements. A sensor at a fixed wall may report the ball's force on the wall. With motion towards the wall positive, that force is positive. The force on the ball is opposite, so reverse the reported sign before comparing it with the ball's negative momentum change. Confirm the instrument's calibrated sign convention rather than assuming its displayed sign already refers to the ball.

1. Calibrate and zero the force channel. Check its response to a known force in a known direction, and use a range that captures the pulse without clipping.
2. Record the whole contact interval. Include the start, changing force, peak and return to the baseline. Use actual sample times and a sampling interval short enough to resolve the pulse.
3. Measure the same body's mass and velocities. Use signed velocities just before and just after the contact, so the momentum change and force area refer to the same event.
4. Compare quantities with units and uncertainty. Sum signed force-time areas in N s and compare with m(v - u) in kg m/s. Include or justify neglecting any other impulse in that direction.

Sparse samples can miss a negative peak and underestimate the impulse magnitude. A force-zero shift changes the area throughout the interval. For example, an uncorrected +10 N offset lasting 0.060 s adds +0.60 N s, making this -4.80 N s pulse appear as -4.20 N s. Repeating the same wrongly zeroed measurement does not remove that bias.

A supplied straight-segment model gives an exact geometric area. A sum formed from samples of a real curved pulse is an estimate whose quality depends on timing, response and sampling. The three points below describe the model, not an actual sensor recording.