Lesson 2: Impulse and Momentum

Objectives:

When you have completed this lesson and the homework, you will be able to:

  • define the term impulse
  • understand the relationship between impulse and the change in momentum
  • be able to calculate the impulse from a F vs. t graph, as well as in the special case where the force is constant
  • be able to solve simple momentum-impulse problems

In this lesson we will address the following standard from the Indiana Academic Standards for Physics I: 

P.1.15

Distinguish between the concepts of momentum (using the formula p = mv) and energy.

 

The Lesson

Introduction

From the last lesson, learned that momentum is the product of the mass and the velocity. It is represented by the variable p, and can be expressed as

In addition, we defined a quantity called the "change in the momentum", which tells us how much the momentum of an object increased or decreased. It can be expressed as

In this lesson, we will answer the question "What causes the momentum to change?".

Defining Impulse

If a tennis ball is cruising along with a velocity of +14 m/s, what could possibly cause it to reverse it's direction and head the other way?  Newton's First Law tells us that objects don't normally do this: an outside force is required!   With a tennis ball, it is not unreasonable to assume that the outside force that causes the tennis ball to reverse its direction is a tennis racquet.  The racquet applies a force on the ball, and this force causes the velocity of the tennis ball to change.    A bigger force, the more the velocity will change.

But is it just a force that causes the momentum of a tennis ball to change? Could there be something else involved?   Think about this: what would happen if you applied a force of 100 N on a person who is resting on a skateboard.  Their velocity would change. But what if you applied that force for 1 second? 2 seconds? 10 seconds?   The time would matter!  The longer you applied the force, the more the momentum will change!   

From simple arguments like this, we can develop our next concept:  the combination of an external force, and some idea about how long it acts, will be called impulse.  It is this impulse that causes the momentum to change.  

But how do we find the impulse?  That is a bit of a tricky matter, but there is a special case that is very simple and which we will use for our homework.

A tennis ball being hit. The force on the ball is not constant.  
Shown here is the effect of the force on the ball when the force is at its maximum value

Suppose I make a graph of the force of my tennis racquet on the tennis ball  vs. time, as shown below.  As the ball starts to contact the racquet, the force of the racquet starts to increase. The force keeps increasing until it reaches a maximum of 6 N, and then the force starts to decrease as the ball leaves the racquet. Eventually, after 0.3 sec, the ball is no longer in contact with the racquet and the force is back to zero.

 

This graph shows the force and how long it acts, but how do we get the impulse from this?  The answer is simple: find the area under the graph. The reason is more obscure, and calculus is needed to explain why this would be the appropriate way to find the impulse. But for this class we will just state the following without trying to explain why:

If the force on an object is graphed on the vertical axis, and the time on the horizontal axis, then the area under the curve of the graph is the impulse.  

In our case, this would be the shaded region, shown like this:

To find the area, we can just find the area of a triangle. The height of the triangle is 6 N, and the base is 0.3 sec. 

Since the area under the curve is the impulse, then the impulse is 0.9 N-s.  

A Special Case

Suppose we have a case where the force is constant - which means that it does not change during the course of the interaction. For example, consider a dad pushing his son on a bike.  In this case, our graph might look like this:

The area under the Force vs. Time graph is now a rectangle, so finding the area is simple:

To find the area, we just use

In this special case, when the force is constant, we can find the area by taking the product of the force (the height of the rectangle) and the time (the width).

So, to sum up this lesson so far:

  • The impulse is the area under the curve of a Force vs. Time graph. 
  • In the special case where the force is constant, we can simplify this and write:

 

Momentum and Impulse

And exactly why have we spent all this time looking at the concept of impulse?  Here is why:

An impulse causes a change in momentum

If we know the impulse on a tennis ball, then we know how much the momentum has changed. Mathematically, we can express this relationship like this:

So if we know the impulse, we also know how much the momentum has changed.  

Do the units work out? Are the units of impulse (N-s) the same as the units of momentum change (kg-m/s) ?

So the units are the same!

Example Problems:

Let's solve two example problems to illustrate this principle:

Problem: A dad pushes on his son (who is on a bike) with a constant force of 40 N for a total of 5 seconds. The boy and the bike have a combined mass of 40 kg.  If the bike was initially at rest, what is the final speed of the bike?  
Solution: You may recognize that this is the problem we just used. The graph of the force vs. time looks like this:

And we already determined that the impulse was 200 N-s.  Our Impulse-momentum Theorem tells us that the impulse (200 N-s) equals the change in momentum, so we have the relationship

200 N-s = Dp.

We will now solve this for the final velocity, since we know the mass (40 kg) and the initial velocity (0 m/s).

So the final velocity of the bike must be 5 m/s in order for the momentum change to be equal to the impulse.

Another (more complicated) example

Problem: A 0.05 kg. tennis ball is moving to the left at 5 m/s when it is hit by a tennis racquet.  The force vs. time graph for this collision is shown to the right.  After the collision with the racquet, the ball is moving to the right with what velocity?.  
Solution: We will use the fact that the impulse equals the change in momentum. So, to solve this problem we need to do two steps: first find the impulse, which is the area under the curve and then once we know the impulse, we can find the final velocity so that the momentum change is equal to the impulse.

From the example above, we have already calculated the impulse:

So now we know the impulse, I.  We now use the impulse-momentum theorem to find the final velocity:

So the final velocity of the tennis ball, after the collision, is 13 m/s to the right

 

Summary

  • The change in the momentum tells us how much the momentum of an object increased or decreased. It can be expressed as

  • Impulse is a combination of the force that acts on an object and the time during which it acts.

  • An impulse is what changes the momentum of an object. 

  • The impulse is the area under the curve of a Force vs. Time graph. 
  • In the special case where the force is constant, the impulse is just FDt

  • We can solve problems using the momentum-impulse theorem:

 

 

Links for further study

Need an alternate explanation? Try this site: 
http://www.physicsclassroom.com/Class/momentum/U4L1a.html

 


This is the end of the lesson. You can return to the unit main page or start the homework assignment

 

PhysicsOnline
(c) 2003 Mark Blachly
Indiana Online Academy
 

This page was last updated
03.11.2004