Impulse, momentum and impact

Work and energy integrated the second law over distance. Integrating it over time gives the third great tool of particle kinetics, the principle of impulse and momentum. It is the natural method when forces act for a known time, when they are huge but brief and hard to measure, as in collisions, or when several bodies push on each other and the internal forces are unknown. A paddle diverter kicking a tote sideways, a shuttle bumping into another at a buffer, a parcel bouncing off a guide rail and a package hitting the floor in a drop test are all impulse problems. In each, the details of the force during contact are messy, but the change in momentum is simple.

Linear momentum and impulse

The linear momentum of a particle is \(\mathbf{L} = m\mathbf{v}\), a vector in the direction of the velocity. Because the mass of a particle is constant, the second law can be written as \(\sum\mathbf{F} = d(m\mathbf{v})/dt\). Multiplying by \(dt\) and integrating from time \(t_1\) to \(t_2\) gives

The integral \(\int \mathbf{F}\,dt\) is the linear impulse of the force, measured in newton-seconds or pound-seconds. The principle says that the initial momentum plus the impulses of all forces equals the final momentum. It is a vector equation, so in a plane it gives two scalar equations, one for each component. That is an advantage over work and energy, which gives only one, and it is why impulse-momentum can find directions.

On a graph of force against time, the impulse is the area under the curve. Two very different force histories, a small force acting for a long time and a large force acting briefly, can have the same area and therefore produce the same change in velocity. That is the principle behind every cushioned landing: an airbag, a crumple zone or the foam in a parcel does not reduce the momentum that must be removed, but by stretching the stopping time it reduces the force needed to remove it.

Average force and impulsive forces

When the force history is unknown but its duration can be estimated, the average force over the contact time is \(F_{avg} = \Delta(mv)/\Delta t\). This is the standard way to estimate loads in short events: kicks, impacts, catches and the action of fast actuators. The peak force is always higher than the average. For a force pulse shaped like half a sine wave, the peak is \(\pi/2\), about 1.57, times the average; for a sharp triangular spike it is twice the average.

During a very short impact, forces such as the contact force become so large that ordinary forces, such as the weight of the body or the friction from a conveyor bed, contribute negligible impulse over the few milliseconds of contact. Impulse-momentum analyses of collisions therefore usually include only the impulsive forces and neglect the others during the contact itself. Before and after the impact, of course, the ordinary forces are back in charge.

Conservation of linear momentum

When two or more particles interact, the forces between them are equal and opposite by Newton's third law and act for the same time, so their impulses cancel when the particles are considered together. If no external impulse acts on the group in some direction, or the external impulses are negligible, the total momentum in that direction is conserved:

Conservation of momentum is often the only way to solve a problem in which the internal forces are unknowable, such as a collision, a coupling or a separation. It holds even though kinetic energy is usually lost in such events, to deformation, heat and sound. A common error is to assume that energy is conserved in a collision because momentum is. The two principles are independent: momentum comes from the third law and is conserved whenever external impulses are negligible; mechanical energy is conserved only when no non-conservative forces do work, which is rarely true in an impact.

Impact

An impact is a collision between two bodies in which large forces act for a very short time. If the bodies' centers of mass both lie on the line of impact, which is the common normal to the contact surfaces, the impact is central; otherwise it is eccentric, and rotation becomes involved (Lesson 11). If the velocities are along the line of impact, the impact is direct; if not, it is oblique.

During the impact there are two phases. In the deformation phase, the bodies squeeze together until they have the same velocity along the line of impact. In the restitution phase, they spring apart, more or less completely depending on the materials. The coefficient of restitution measures how complete the rebound is:

That is, \(e\) is the ratio of the speed of separation after impact to the speed of approach before it, both measured along the line of impact. A perfectly elastic impact has \(e = 1\) and loses no energy. A perfectly plastic impact has \(e = 0\): the bodies stick together. Real values depend on the materials, the impact speed and the shapes. Hardened steel balls may have \(e\) around 0.9; rubber buffers perhaps 0.5 to 0.8; wooden pallets and filled cartons are often well below 0.5.

For a direct central impact, conservation of momentum and the definition of \(e\) give two equations in the two unknown final velocities. Solving them in general:

It is usually clearer to write the two equations and solve them for each problem than to memorize these formulas, but they make good checks.

Oblique impact

In an oblique impact, split each velocity into components along the line of impact (the normal) and along the contact plane (the tangent). If the contact surfaces are smooth, there is no tangential impulse, so each body's tangential velocity component is unchanged. Along the normal, conservation of momentum and the coefficient of restitution apply exactly as in a direct impact. When one body is fixed, such as a wall or a guide rail, only restitution applies along the normal: the rebounding normal velocity is \(e\) times the approaching one.

Friction between the parcel and the guide during contact would also reduce the tangential component, and a rotating parcel would complicate matters further. For layout purposes, the smooth-contact model is usually adequate; for fragile items, guide liners with low friction and soft facings are chosen to keep both the normal impulse and the tangential drag small.

Measuring an impact: the drop test

Package designers test their designs by dropping them from set heights and measuring how hard they land. The coefficient of restitution can be found from the drop and rebound heights alone, because the impact and rebound speeds come from energy conservation during the free fall: \(e = \sqrt{h_{up}/h_{drop}}\). The impulse is the change in momentum, and a load cell or accelerometer gives the duration and shape of the pulse. The program below works through a 4.0 kg package dropped from 0.76 m that rebounds to 0.12 m, with a measured contact time of 6 ms.

# Drop test: a 4.0 kg package falls 0.76 m onto a rigid floor and
# rebounds 0.12 m. A load cell under the floor plate records a roughly
# half-sine force pulse lasting 6 ms. Find e, the impulse, and the forces.
import math

g, m = 9.81, 4.0
h_drop, h_up, tau = 0.76, 0.12, 0.006

v_in = math.sqrt(2 * g * h_drop)          # speed just before impact
v_out = math.sqrt(2 * g * h_up)           # speed just after
e = v_out / v_in
J = m * (v_in + v_out)                    # impulse (velocity reverses)
print(f"impact {v_in:.3f} m/s, rebound {v_out:.3f} m/s, e = {e:.3f}")
print(f"impulse from momentum change: {J:.2f} N s")

# Build the sampled pulse a load cell would record (50 kHz) with the
# peak chosen so its area equals J, then integrate it like real data.
F_peak = J * math.pi / (2 * tau)
dt = 1 / 50000
samples = [F_peak * math.sin(math.pi * k * dt / tau) for k in range(int(tau / dt) + 1)]
area = sum((samples[k] + samples[k + 1]) / 2 * dt for k in range(len(samples) - 1))
print(f"integrated pulse area: {area:.2f} N s")
print(f"average force {J / tau:7.0f} N   peak force {F_peak:7.0f} N")
print(f"peak deceleration {F_peak / m / g:.0f} g")
print(f"kinetic energy lost: {0.5 * m * (v_in**2 - v_out**2):.1f} J of {0.5 * m * v_in**2:.1f} J")

impact 3.862 m/s, rebound 1.534 m/s, e = 0.397
impulse from momentum change: 21.58 N s
integrated pulse area: 21.58 N s
average force    3597 N   peak force    5651 N
peak deceleration 144 g
kinetic energy lost: 25.1 J of 29.8 J

The numbers are sobering. A drop of under a meter gives an average contact force of about 3,600 N on a 4 kg package, close to a hundred times its weight, and a peak near 144 g. The package's weight, about 39 N, contributes less than 1 percent of the impulse during the 6 ms contact, which justifies leaving it out. The cushioning inside the box exists to lengthen the contact time for the product itself: if the foam stretches the product's stopping time from 6 ms to 30 ms, the peak force on the product falls fivefold for the same impulse.