Linear Momentum, Force, and Impulse, Momentum | Physics

Learning objectives

Section Key Terms

change in momentum

impulse

impulse–momentum theorem

linear momentum

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

p = m v .

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

F_{net} = \frac{Δ p}{Δ t},

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

We can solve for $Δ p$ by rearranging the equation

F_{net} = \frac{Δ p}{Δ t}

to be

Δ p = F_{net} Δ t .

$F_{net} Δ t$ 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 $Δ p = Δ (m v)$ . 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. $F_{net} = m a$ is actually derived from the equation:

F_{net} = \frac{Δ p}{Δ t}

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

F_{net} = \frac{Δ p}{Δ t}

by substituting the definitions of acceleration and momentum.

The change in momentum $Δ p$ is given by

Δ p = Δ (m v) .

If the mass of the system is constant, then

Δ (m v) = m Δ v .

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

F_{net} = \frac{Δ p}{Δ t} = \frac{m Δ v}{Δ t}

for a constant mass.

Because

\frac{Δ v}{Δ t} = a,

we can substitute to get the familiar equation

F_{net} = m a

when the mass of the system is constant.

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_08_01_Airbag). 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 $Δ p = F_{net} Δ t$ .

View original figure (external; image not reproduced in this text edition)

Original figure description: An illustration is shown of a man sitting in a car seat behind the wheel, wearing a seat belt. An airbag has been deployed from the wheel. The illustration indicates that the man's body is moving toward the airbag.

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 $F_{net} = \frac{Δ p}{Δ t},$ you can see how increasing $Δ t$ while $Δ p$ 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.

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.

Bending your knees increases the time of the impact, thus decreasing the force.
Bending your knees decreases the time of the impact, thus decreasing the force.
Bending your knees increases the time of the impact, thus increasing the force.
Bending your knees decreases the time of the impact, thus increasing the force.

Solving Problems Using the Impulse-Momentum Theorem

Calculating Momentum: A Football Player and a Football

(a) Calculate the momentum of a 110 kg football player running at 8 m/s. (b) Compare the player’s momentum with the momentum of a 0.410 kg football thrown hard at a speed of 25 m/s.

Solution for (a)

To find the player’s momentum, substitute the known values for the player’s mass and speed into the equation.

p_{player} = (110 kg) (8 m/s) = 880 k g \cdot m / s

Solution for (b)

To find the ball’s momentum, substitute the known values for the ball’s mass and speed into the equation.

p_{ball} = (0.410 kg) (25 m/s) = 10.25 k g \cdot m/s

The ratio of the player’s momentum to the ball’s momentum is

\frac{p_{player}}{p_{ball}} = \frac{880}{10.3} = 85.9 .

Discussion

Although the ball has greater velocity, the player has a much greater mass. Therefore, the momentum of the player is about 86 times greater than the momentum of the football.

Calculating Force: Venus Williams’ Racquet

During the 2007 French Open, Venus Williams (Figure_08_01_Williams) 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).

View original figure (external; image not reproduced in this text edition)

Original figure description: Serena Williams is a famous tennis player. In this image she is hitting a tennis ball with a tennis racket.

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

Solution

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

\begin{matrix} Δ p & = & m (v_{f} - v_{i}) \\ = & (0 .057 kg) (58 m/s – 0 m/s) \\ = & 3 .306 kg·m/s \approx 3 .3 kg·m/s \end{matrix}

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

\begin{matrix} F_{net} & = & \frac{Δ p}{Δ t} = \frac{3.306}{5 \times 10^{- 3}} \\ = & 661 N \\ \approx & 660 N. \end{matrix}

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.

Practice Problems

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Check Your Understanding

What is linear momentum?

the sum of a systems mass and its velocity
the ratio of a systems mass to its velocity
the product of a systems mass and its velocity
the product of a systems moment of inertia and its velocity

If an objects mass is constant, what is its momentum proportional to?

Its velocity
Its weight
Its displacement
Its moment of inertia

What is the equation for Newtons second law of motion, in terms of mass, velocity, and time, when the mass of the system is constant?

$F_{net} = \frac{Δ v}{Δ m Δ t}$
$F_{net} = \frac{m Δ t}{Δ v}$
$F_{net} = \frac{m Δ v}{Δ t}$
$F_{net} = \frac{Δ m Δ v}{Δ t}$

Give an example of a system whose mass is not constant.

A spinning top
A baseball flying through the air
A rocket launched from Earth
A block sliding on a frictionless inclined plane

Section Summary

Linear momentum, often referenced as momentum for short, is defined as the product of a system’s mass multiplied by its velocity, p = mv.
The SI unit for momentum is kg m/s.
Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes, $F_{net} = \frac{Δ p}{Δ t}$ .
Impulse is the average net external force multiplied by the time this force acts, and impulse equals the change in momentum, $Δ p = F_{net} Δ t$ .
Forces are usually not constant over a period of time, so we use the average of the force over the time it acts.

Key Equations

impulse	$F_{net} Δ t$
impulse–momentum theorem	$Δ p = F_{net} Δ t$
linear momentum	$p = m v$
Newton’s second law in terms of momentum	$F_{net} = \frac{Δ p}{Δ t}$

Concept Items

What is impulse?

Change in velocity
Change in momentum
Rate of change of velocity
Rate of change of momentum

In which equation of Newtons second law is mass assumed to be constant?

$F = m a$
$F = \frac{Δ p}{Δ t}$
$F = Δ p Δ t$
$F = \frac{Δ m}{Δ a}$

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What is the equation for linear momentum?

$p = m v$
$p = m / v$
$p = m v^{2}$
$p = \frac{1}{2} m v^{2}$

Critical Thinking

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Cars these days have parts that can crumple or collapse in the event of an accident. How does this help protect the passengers?

It reduces injury to the passengers by increasing the time of impact.
It reduces injury to the passengers by decreasing the time of impact.
It reduces injury to the passengers by increasing the change in momentum.
It reduces injury to the passengers by decreasing the change in momentum.

How much force would be needed to cause a 17 kg ⋅ m/s change in the momentum of an object, if the force acted for 5 seconds?

3.4 N
12 N
22 N
85 N

Problems

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For how long should a force of 130 N be applied to an object of mass 50 kg to change its speed from 20 m/s to 60 m/s?

0.031 s
0.065 s
15.4 s
40 s

Test Prep Multiple Choice

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When does the net force on an object increase?

When Δp decreases
When Δt increases
When Δt decreases

In the equation Δp = m(v_f − v_i), which quantity is considered to be constant?

Initial velocity
Final velocity
Mass
Momentum

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Test Prep Short Answer

If an objects velocity is constant, what is its momentum proportional to?

Its shape
Its mass
Its length
Its breadth

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How can you express impulse in terms of mass and velocity when neither of those are constant?

$Δ p = Δ (m v)$
$\frac{Δ p}{Δ t} = \frac{Δ (m v)}{Δ t}$
$Δ p = Δ (\frac{m}{v})$
$\frac{Δ p}{Δ t} = \frac{1}{Δ t} · Δ (m v)$

How can you express impulse in terms of mass and initial and final velocities?

$Δ p = m (v_{f} - v_{i})$
$\frac{Δ p}{Δ t} = \frac{m (v_{f} - v_{i})}{Δ t}$
$Δ p = \frac{(v_{f} - v_{i})}{m}$
$\frac{Δ p}{Δ t} = \frac{1}{m} \frac{(v_{f} - v_{i})}{Δ t}$

Why do we use average force while solving momentum problems? How is net force related to the momentum of the object?

Forces are usually constant over a period of time, and net force acting on the object is equal to the rate of change of the momentum.
Forces are usually not constant over a period of time, and net force acting on the object is equal to the product of the momentum and the time interval.
Forces are usually constant over a period of time, and net force acting on the object is equal to the product of the momentum and the time interval.
Forces are usually not constant over a period of time, and net force acting on the object is equal to the rate of change of the momentum.

Test Prep Extended Response

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Can we use the equation $F_{net} = \frac{Δ p}{Δ t}$ when the mass is constant?

No, because the given equation is applicable for the variable mass only.
No, because the given equation is not applicable for the constant mass.
Yes, and the resultant equation is F = mv
Yes, and the resultant equation is F = ma

Glossary

change in momentum: the difference between the final and initial values of momentum; the mass times the change in velocity
impulse: average net external force multiplied by the time the force acts; equal to the change in momentum
impulse–momentum theorem: the impulse, or change in momentum, is the product of the net external force and the time over which the force acts
linear momentum: the product of a system's mass and velocity

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