Impulse
The time integral of force. The net external impulse acting on a system over a given time interval is equal to the system's change in momentum.
Motivation for Concept
Although some everyday interactions like gravity and friction result in stable forces whose effects can easily be analyzed with dynamics, many interactions are not steady. Consider, for example, the difference between a push and a punch. When you push something, you consciously use your muscles to apply a steady force to the target object. For this reason, pushing a bowling ball or a bean bag feels much the same, apart from the fact that the bean bag will probably deform more in response to the force. When you punch something, however, you simply let your fist fly. The force felt by the target object and the reaction force exterted on your fist are the result of the impact of your moving fist with the object. This impact is essentially out of your control. The force exterted are not determined directly by your muscles (though the faster your fist is moving, the greater the force will tend to be), but rather it is principally determined by the properties of your hand and the target object. Because of this fact, there is a dramatic difference between the results of punching a bean bag versus punching a bowling ball.
Such impact or collision forces are extremely common in everyday life. Almost any sport will involve collisions. Household activities like hammering nails or kneading dough require collisions. Understanding collisions is also of great importance to car manufacturers.
Unfortunately, the forces during a collision are very difficult to characterize. They change extremely rapidly in time (the entire duration of a typical collision is measured in milliseconds), and they manner in which they change is stronly dependent on the material properties of the objects undergoing the collision. Because of these complications, it is rare to see a detailed force profile for a collision. Instead, collisions are usually described by an effect that is much more easily observed: the resulting change in the motion of the participants. The impulse delivered by a collision is one measure of this change.
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Photos taken with a high-speed digital camera showing the impact of a (new) tennis ball dropped from a height of 100 inches onto a wooden platform. |
Note that the center images are less blurry than the first and last. Can you explain this?
The ball rebounds because it is elastic. The deformation of the ball observed in the center image is not permanent. The ball's structure causes it to return to its initial shape, and in the process it pushes itself off the ground. If the ball were made of clay, the deformation would remain and the ball would simply "splat" onto the ground (one more example of how structural differences complicate collision analysis).
Definition of Net Impulse in terms of Momentum
Vector Change in Momentum
Suppose an object experiences a sudden interaction that results in a dramatic change in the object's momentum. One definition of the net impulse provided during the change in motion is to calculate the numerical value of the change in the object's momentum. In other words, the impulse J is:
\begin
[ \vec
_
= \vec
_
- \vec
_
= m\vec
_
- m\vec
_
]\end
Note that because momentum is a vector, the change in momentum is also a vector. Thus, the impulse is by defintion a vector quantity. As with any vector quantity, it is important to remember that the calculation of impulse really involves three equations, one for each component:
\begin
[ J_{{\rm net},x} = mv_
- mv_
]
[ J_{{\rm net},y} = mv_
- mv_
]
[ J_{{\rm net},z} = mv_
- mv_
]\end
Initial-State Final-State Diagrams
The vector nature of momentum means that it very important when calculating the change to carefully set up a coordinate system. For this reason, it is strongly recommended that you begin any problem involving a change in momentum by constructing an intial-state final-state diagram.
Consider, for example, a ball of mass mb that is moving to the right at a constant speed vb when it suddenly impacts a wall and reverses direction (still moving at the same speed). The magnitude of the momentum before and after the collsion is the same (mbvb), which can easily lead to the conclusion that there has been no change. Thinking about the situation, however, should quickly convince you that the ball has certainly been acted on by some force, which implies that a change did occur. Carefully drawing the diagrams below (taking special note of the coordinate system) shows the resolution to this difficulty.
Unable to render embedded object: File (ballreversei.png) not found. |
Unable to render embedded object: File (ballreversef.png) not found. |
Initial State |
Final State |
|---|
The ball's initial x momentum is positive in our coordinates (+mbvb), while its final x momentum is negative ( – mbvb), giving a change of:
\begin
[ J_
= -m_
v_
- m_
v_
= -2m_
v_
]\end
where the negative sign indicates that the impulse is applied in the negative x direction, and so the impulse points leftward in this case.
FindingImpulseusingVectorDiagrams"> Finding Impulse using Vector Diagrams
Finding the change in a vector quantity like momentum can be very confusing. Here are two possible ways to think about the role of impulse in a vector diagram.
Adding to get the Final Momentum "> Adding to get the Final Momentum
We have defined impulse as the final momentum minus the initial momentum, but subtracting vectors can be confusing. Therefore, we will first consider a rearrangement of the definition of impulse. We can write:
\begin
[ \vec
_
= \vec
_
+ \vec
] \end
Thus, we can consider the impulse as the vector we must add to the initial momentum to yield the final momentum.
We can use this formulation to draw a vector diagram representing the ball-wall collision described in the previous section (Initial-State Final-State Diagrams). Remembering the rules for adding vectors tail-to-tip, we can draw the following diagram which includes the impulse vector:
Unable to render embedded object: File (addimpulsevec.png) not found.
Subtracting Initial Momentum from Final">Subtracting Initial Momentum from Final
It is also possible to draw a vector representation of the regular definition of impulse
\begin
[ \vec
= \vec
_
- \vec
_
] \end
but drawing a vector equation that includes subtraction is tricky. We must think of this equation in the following way:
\begin
[ \vec
= \vec
_
+ (- \vec
_
) ] \end
In other words, we must think of the right hand side as the final momentum plus the negative of the initial momentum vector. Since the negative of a vector is just the reversed vector, this leads to the picture:
Unable to render embedded object: File (addneginitvec.png) not found.
which gives the same impulse vector as the diagram above.
ForceandImpulse"> Force and Impulse
ImpulseandNewton'sSecondLaw"> Impulse and Newton's Second Law
The integral form of Newton's 2nd Law states that:
\begin
[ \int_{t_{i}}^{t_{f}} \vec
_
\;dt = p_
- p_
]\end
Comparing this with our first definition of impulse, it is obvious that we can also define the total impulse as:
\begin
[ \vec
_
= \int_{t_{i}}^{t_{f}} \vec
_
\; dt ]\end
Inspired by this definition, we will sometimes refer to the impulse provided by a single force as:
\begin
[ \vec
= \int_{t_{i}}^{t_{f}} \vec
\; dt ]\end
so that the net impulse acting on a system is:
\begin
[ \vec
_
= \sum_
\vec
]\end
Impulses resulting from internal forces cancel by Newton's Third Law.
Time-AveragedForce"> Time-Averaged Force
Since collision forces often have a complicated time evolution, it is rare for them to be easily integrable as a function of time. Thus, it is rare for the definition of impulse in terms of force to be used to find impulse. Instead, it is often used "in reverse" to characterize the forces involved in a collision. First, the change in impulse of the object involved in the collision is determined, then the impulse is used to estimate the force. Again, the detailed form of the force will be too complicated to find in this way, but if the total duration of the collision, Δt=tf - ti, is known, then it is possible to find the time-averaged net force experienced by the object during the collision. The time-averaged net force is, by definition:
\begin
[ <\vec
_
>_
\equiv \frac{\int_{t_{i}}^{t_{f}} \vec
_
dt}{t_
-t_{i}}]\end
This definition makes it clear that the time-averaged net force is directly related to the net impulse:
\begin
[ <\vec
_
>_
= \frac{\vec
{\rm net}}{t
-t_{i}} = \frac{\vec
_{\rm net}}
]\end
To see why different averages need not give the same value, consider two babysitters who each make $15 per hour. They have the same time-averaged income. Suppose, however, that one of the sitters gets jobs that last 3 hours on average, while the other's jobs are 4 hours. Their job-averaged income is different ($45 vs. $60). Similarly, you could imagine two people who mow lawns for extra income, charging $40 per lawn. Their job-averaged income is the same, but their time-averaged income could be different if they tend to mow different sized lawns.
ExampleProblemsUsingTime-AveragedForce"> Example Problems Using Time-Averaged Force
AverageForceandtheDirectionofImpulse"> Average Force and the Direction of Impulse
Remembering that the impulse is directly proportional to the average net force in a collision may help you to gain an intuitive understanding for the direction of the impulse in many problems. Because they are directly related, the net impulse must point in the same direction as the average net force. With this principle in mind, look back at the example of the ball-wall collision given above, and consider again why the impulse must point to the left in this collision.