Momentum, Impulse, and the Impulse-Momentum Theorem

Linear momentum is the product of a system’s mass and its velocity. In equation form, linear momentum p is

You can see from the equation that momentum is directly proportional to the object’s mass (m) and velocity (v). Therefore, the greater an object’s mass or the greater its velocity, the greater its momentum. A large, fast-moving object has greater momentum than a smaller, slower object.

Momentum is a vector and has the same direction as velocity v. Since mass is a scalar, when velocity is in a negative direction (i.e., opposite the direction of motion), the momentum will also be in a negative direction; and when velocity is in a positive direction, momentum will likewise be in a positive direction. The SI unit for momentum is kg m/s.

Momentum is so important for understanding motion that it was called the quantity of motion by physicists such as Newton. Force influences momentum, and we can rearrange Newton’s second law of motion to show the relationship between force and momentum.

Recall our study of Newton’s second law of motion (F_net = ma). Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. The change in momentum is the difference between the final and initial values of momentum.

In equation form, this law is

where F_net is the net external force, $\text{Δ}p$ is the change in momentum, and $\text{Δ}t$ is the change in time.

We can solve for $\text{Δ}p$ by rearranging the equation

to be

${\text{F}}_{\text{net}}\text{Δ}t$ is known as impulse and this equation is known as the impulse-momentum theorem. From the equation, we see that the impulse equals the average net external force multiplied by the time this force acts. It is equal to the change in momentum. The effect of a force on an object depends on how long it acts, as well as the strength of the force. Impulse is a useful concept because it quantifies the effect of a force. A very large force acting for a short time can have a great effect on the momentum of an object, such as the force of a racket hitting a tennis ball. A small force could cause the same change in momentum, but it would have to act for a much longer time.

Newton’s Second Law in Terms of Momentum

When Newton’s second law is expressed in terms of momentum, it can be used for solving problems where mass varies, since $\text{Δ}\text{p}=\text{Δ}(m\text{v})$. In the more traditional form of the law that you are used to working with, mass is assumed to be constant. In fact, this traditional form is a special case of the law, where mass is constant.${\text{F}}_{\text{net}}=m\text{a}$ is actually derived from the equation:

For the sake of understanding the relationship between Newton’s second law in its two forms, let’s recreate the derivation of ${\text{F}}_{\text{net}}=m\text{a}$ from

by substituting the definitions of acceleration and momentum.

The change in momentum $\text{Δ}\text{p}$ is given by

If the mass of the system is constant, then

By substituting $m\text{Δ}\text{v}$ for $\text{Δ}\text{p}$, Newton’s second law of motion becomes

for a constant mass.

Because

we can substitute to get the familiar equation

when the mass of the system is constant.

Links To Physics

Engineering: Saving Lives Using the Concept of Impulse

Cars during the past several decades have gotten much safer. Seat belts play a major role in automobile safety by preventing people from flying into the windshield in the event of a crash. Other safety features, such as airbags, are less visible or obvious, but are also effective at making auto crashes less deadly (see Figure 8.2). Many of these safety features make use of the concept of impulse from physics. Recall that impulse is the net force multiplied by the duration of time of the impact. This was expressed mathematically as $\text{Δ}\text{p}={\text{F}}_{\text{net}}\text{Δ}t$.

Figure 8.2 Vehicles have safety features like airbags and seat belts installed.

Airbags allow the net force on the occupants in the car to act over a much longer time when there is a sudden stop. The momentum change is the same for an occupant whether an airbag is deployed or not. But the force that brings the occupant to a stop will be much less if it acts over a larger time. By rearranging the equation for impulse to solve for force $F_{\text{net}}=\frac{\Delta p}{\Delta t},$ you can see how increasing $\text{Δ}t$ while $\text{Δ}p$ stays the same will decrease F_net. This is another example of an inverse relationship. Similarly, a padded dashboard increases the time over which the force of impact acts, thereby reducing the force of impact.

Cars today have many plastic components. One advantage of plastics is their lighter weight, which results in better gas mileage. Another advantage is that a car will crumple in a collision, especially in the event of a head-on collision. A longer collision time means the force on the occupants of the car will be less. Deaths during car races decreased dramatically when the rigid frames of racing cars were replaced with parts that could crumple or collapse in the event of an accident.

Grasp Check

You may have heard the advice to bend your knees when jumping. In this example, a friend dares you to jump off of a park bench onto the ground without bending your knees. You, of course, refuse. Explain to your friend why this would be a foolish thing. Show it using the impulse-momentum theorem.

1. Bending your knees increases the time of the impact, thus decreasing the force.
2. Bending your knees decreases the time of the impact, thus decreasing the force.
3. Bending your knees increases the time of the impact, thus increasing the force.
4. Bending your knees decreases the time of the impact, thus increasing the force.

Solving Problems Using the Impulse-Momentum Theorem

Worked Example

Calculating Force: Venus Williams’ Racquet

During the 2007 French Open, Venus Williams (Figure 8.3) hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What was the average force exerted on the 0.057 kg tennis ball by Williams’ racquet? Assume that the ball’s speed just after impact was 58 m/s, the horizontal velocity before impact is negligible, and that the ball remained in contact with the racquet for 5 ms (milliseconds).

Figure 8.3 Venus Williams playing in the 2013 US Open (Edwin Martinez, Flickr)

Strategy

Recall that Newton’s second law stated in terms of momentum is

$$F_{\text{net}}=\frac{\text{Δ}p}{\text{Δ}t}\text{.}$$

As noted above, when mass is constant, the change in momentum is given by

$$\text{Δ}\text{p}=m\text{Δ}\text{v}=m({\text{v}}_{\text{f}}-{\text{v}}_{\text{i}})\text{,}$$

where v_f is the final velocity and v_i is the initial velocity. In this example, the velocity just after impact and the change in time are given, so after we solve for $\text{Δ}\text{p}$, we can use $F_{\text{net}}=\frac{\text{Δ}p}{\text{Δ}t}$ to find the force.

Solution

To determine the change in momentum, substitute the values for mass and the initial and final velocities into the equation above.

8.1$$\begin{array}{ccc}\hfill \text{Δ}\text{p}& =& m({\text{v}}_f-{\text{v}}_i)\hfill \\ & =& \left(\text{0}\text{.057 kg}\right)\left(\text{58 m/s – 0 m/s}\right)\hfill \\ & =& \text{3}\text{.306 kg·m/s }\approx \text{ 3}\text{.3 kg·m/s}\hfill \end{array}$$

Now we can find the magnitude of the net external force using $F_{\text{net}}=\frac{\text{Δ}p}{\text{Δ}t}$

8.2$$\begin{array}{ccc}\hfill F_{\text{net}}& =& \frac{\text{Δ}p}{\text{Δ}t}=\frac{3.306}{5\times {10}^{-3}}\hfill \\ & =& 661\text{ N}\hfill \\ & \approx & 660\text{ N.}\hfill \end{array}$$

Discussion

This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact. This problem could also be solved by first finding the acceleration and then using F_net = ma, but we would have had to do one more step. In this case, using momentum was a shortcut.