11.1 Momentum and Impulse

A collision is a short-duration interaction between two objects that strike each other. The collision between a tennis ball and a racket, or a baseball and a bat, may seem instantaneous to your eye, but that is a limitation of your perception. A high-speed photograph reveals that the side of the ball is significantly flattened during the collision. It takes time to compress the ball, and more time for the ball to re-expand as it leaves the racket or bat.

A photograph shows a tennis ball colliding with a tennis racket. The region where the ball strikes the racket’s net appears to be displaced in the direction of the colliding ball. — A tennis ball collides with a racket. Notice that the right side of the ball is flattened.

The duration of a collision depends on the materials from which the objects are made, but 1 to 10 ms (0.001 to 0.010 s) is fairly typical. This is the time during which the two objects are in contact with each other. The harder the objects, the shorter the contact time. A collision between two steel balls lasts less than 200 microseconds.

FIGURE 11.1 shows an object colliding with a wall. The object approaches with an initial velocity $\vec{v}_{i}$ , experiences a force of duration $Δ t$ , and leaves with final velocity $\vec{v}_{f}$ . Notice that the object, as in the photo above, deforms during the collision. A particle cannot be deformed, so we cannot model colliding objects as particles. Instead, we model a colliding object as an elastic object that compresses and then expands, much like a spring. Indeed, that’s exactly what happens at the microscopic level during a collision: Molecular bonds compress, store elastic potential energy, then transform some or all of that potential energy back into the kinetic energy of the rebounding object. We’ll examine the energy issues of collisions later in this chapter.

The figure consists of two parts and illustrates the collision of a spherical object with a wall.

FIGURE 11.1 Full Alternative Text

The force of a collision is usually very large in comparison to other forces exerted on the object. A large force exerted for a small interval of time is called an impulsive force. The graph of Figure 11.1 shows how a typical impulsive force behaves, rapidly growing to a maximum at the instant of maximum compression, then decreasing back to zero. The force is zero before contact begins and after contact ends. Because an impulsive force is a function of time, we will write it as $\vec{F} (t)$ .

We would like to know how a collision changes an object’s velocity, and we can use Newton’s second law to do that. Acceleration is $\vec{a} = d \vec{v} / d t$ , so the second law is

m \vec{a} = m \frac{d \vec{v}}{d t} = \vec{F} (t)

Multiplying both sides by $d t$ gives

m d \vec{v} = \vec{F} (t) d t

(11.1)

The force differs from zero only during the interval of time from $t_{i}$ to $t_{f} = t_{i} + Δ t$ , so let’s integrate Equation 11.1 over just this interval. The velocity changes from $\vec{v}_{i}$ to $\vec{v}_{f}$ during the collision; thus

m \int_{{\vec{v}}_{i}}^{{\vec{v}}_{f}} d \vec{v} = m {\vec{v}}_{f} - m {\vec{v}}_{i} = \int_{t_{i}}^{t_{f}} \vec{F} (t) d t

(11.2)

We need some new tools to help us make sense of Equation 11.2.