By the end of this section, you will be able to do the following:
- Describe momentum, what can change momentum, impulse, and the impulse-momentum theorem
- Describe Newton’s second law in terms of momentum
- Solve problems using the impulse-momentum theorem
The learning objectives in this section will help your students master the following standards:
- (6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (C) calculate the mechanical energy of, power generated within, impulse applied to, and momentum of a physical system.
Section Key Terms
|change in momentum||impulse||impulse–momentum theorem||linear momentum|
[BL][OL] Review inertia and Newton’s laws of motion.
[AL] Start a discussion about movement and collision. Using the example of football players, point out that both the mass and the velocity of an object are important considerations in determining the impact of collisions. The direction as well as the magnitude of velocity is very important.
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 (Fnet = 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 Fnet is the net external force, is the change in momentum, and is the change in time.
We can solve for by rearranging the equation
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.
[OL][AL] Explain that a large, fast-moving object has greater momentum than a smaller, slower object. This quality is called momentum.
[BL][OL] Review the equation of Newton’s second law of motion. Point out the two different equations for the law.
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 . 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. 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 from
by substituting the definitions of acceleration and momentum.
The change in momentum is given by
If the mass of the system is constant, then
By substituting for , Newton’s second law of motion becomes
for a constant mass.
we can substitute to get the familiar equation
when the mass of the system is constant.
[BL][OL][AL] Show the two different forms of Newton’s second law and how one can be derived from the other.
We just showed how applies only when the mass of the system is constant. An example of when this formula would not apply would be a moving rocket that burns enough fuel to significantly change the mass of the rocket. In this case, you can use Newton’s second law expressed in terms of momentum to account for the changing mass without having to know anything about the interaction force by the fuel on the rocket.
Hand Movement and Impulse
In this activity you will experiment with different types of hand motions to gain an intuitive understanding of the relationship between force, time, and impulse.
- one ball
- one tub filled with water
- Try catching a ball while giving with the ball, pulling your hands toward your body.
- Next, try catching a ball while keeping your hands still.
- Hit water in a tub with your full palm. Your full palm represents a swimmer doing a belly flop.
- After the water has settled, hit the water again by diving your hand with your fingers first into the water. Your diving hand represents a swimmer doing a dive.
- Explain what happens in each case and why.
[OL][AL] Discuss the impact one feels when one falls or jumps. List the factors that affect this impact.
Solving Problems Using the Impulse-Momentum Theorem
Talk about the different strategies to be used while solving problems. Make sure that students know the assumptions made in each equation regarding certain quantities being constant or some quantities being negligible.
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.
No information is given about the direction of the football player or the football, so we can calculate only the magnitude of the momentum, p. (A symbol in italics represents magnitude.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum:
To find the player’s momentum, substitute the known values for the player’s mass and speed into the equation.
To find the ball’s momentum, substitute the known values for the ball’s mass and speed into the equation.
The ratio of the player’s momentum to the ball’s momentum is
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 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).
Recall that Newton’s second law stated in terms of momentum is
As noted above, when mass is constant, the change in momentum is given by
where vf is the final velocity and vi is the initial velocity. In this example, the velocity just after impact and the change in time are given, so after we solve for , we can use to find the force.
To determine the change in momentum, substitute the values for mass and the initial and final velocities into the equation above.
Now we can find the magnitude of the net external force using
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 Fnet = ma, but we would have had to do one more step. In this case, using momentum was a shortcut.
Check Your Understanding
What is linear momentum?
- the sum of a system’s mass and its velocity
- the ratio of a system’s mass to its velocity
- the product of a system’s mass and its velocity
- the product of a system’s moment of inertia and its velocity
If an object’s mass is constant, what is its momentum proportional to?
- Its velocity
- Its weight
- Its displacement
- Its moment of inertia
What is the equation for Newton’s second law of motion, in terms of mass, velocity, and time, when the mass of the system is constant?
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
Use the Check Your Understanding questions to assess whether students master the learning objectives of this section. If students are struggling with a specific objective, the assessment will help identify which objective is causing the problem and direct students to the relevant content.