11.1.2 Impulse

Equation 11.5 tells us that an object’s change in momentum is equal to the time integral of the force. Let’s define that integral to be a quantity $\vec{J}$ called the impulse:

i m p u l s e = \vec{J} \equiv \int_{t_{i}}^{t_{f}} \vec{F} (t) d t

(11.6)

Impulse has units of $N s$ , but you should be able to show that $N s$ is equivalent to $kg m / s$ , the units of momentum.

Thus Equation 11.2, which we found by integrating Newton’s second law, can be rewritten in terms of impulse and momentum as

Δ \vec{p} = \vec{J} (momentum principle)

(11.7)

This result, called the momentum principle, says that an impulse delivered to an object causes the object’s momentum to change. The momentum $\vec{p}_{f}$ “after” an interaction, such as a collision, is equal to the momentum $\vec{p}_{i}$ “before” the interaction plus the impulse that arises from the interaction:

\vec{p}_{f} = \vec{p}_{i} + \vec{J}

(11.8)

For motion in one dimension—say, along the x-axis—the momentum principle is

Δ p_{x} = p_{f x} - p_{i x} = J_{x}

(11.9)

where the x-component of the impulse is

J_{x} = \int_{t_{i}}^{t_{f}} F_{x} (t) d t = area under ​ ​ the F_{x} (t) curve between t_{i} and t_{f}

(11.10)

The interpretation of the integral in Equation 11.10 as an area under a curve is especially important. FIGURE 11.3a portrays the impulse graphically. Because the force changes in a complicated way during a collision, it is often useful to describe the collision in terms of an average force $F_{avg}$ . As Figure 11.3b shows, $F_{avg}$ is the height of a rectangle that has the same area, and thus the same impulse, as the real force curve. The impulse exerted during the collision is

J_{x} = F_{avg} Δ t

(11.11)

The figure consists of two parts labeled (“a”) and (b), respectively. Both parts illustrate impulse graphs for the momentum of a moving particle.

FIGURE 11.3 Full Alternative Text

FIGURE 11.4 illustrates the momentum principle for a rubber ball bouncing off a wall. Notice the signs; they are very important. The ball is initially traveling toward the right, so $v_{i x}$ and $p_{i x}$ are positive. After the bounce, $v_{f x}$ and $p_{f x}$ are negative. The force on the ball is toward the left, so $F_{x}$ is also negative. The graphs show how the force and the momentum change with time.

The figure consists of three parts that illustrate a rubber ball colliding with a wall and the result of the collision and the graphical representations of the rubber ball’s momentum before and after its collision.

FIGURE 11.4 Full Alternative Text

Although the interaction is very complex, the impulse—the area under the force graph—is all we need to know to find the ball’s velocity as it rebounds from the wall. The final momentum is

p_{f x} = p_{i x} + J_{x} = p_{i x} + area under the force curve

(11.12)

and the final velocity is $v_{f x} = p_{f x} / m$ . In this example, the area has a negative value.

11.1.2 Impulse

FIGURE 11.3 Looking at the impulse graphically.

FIGURE 11.4 The momentum principle helps us understand a rubber ball bouncing off a wall.