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.
MotivationforConcept"> Motivation for Concept
OneDefinitionofNetImpulse"> One Definition of Net Impulse
VectorChangeinMomentum"> 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-StateFinal-StateDiagrams"> 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
Newton'sSecondLaw"> 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{large]
Impulses resulting from internal forces cancel by Newton's Third Law.
Time-AveragedForce"> Time-Averaged Force
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.