Lesson 2: Impulse and Momentum
When you have completed this lesson and the homework, you will be able to:
In this lesson we will address the following standard from the Indiana Academic Standards for Physics I:
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?".
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
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:
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:
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!
Let's solve two example problems to illustrate this principle:
Another (more complicated) example
Links for further studyNeed an alternate explanation? Try this site: