Resource Lesson
A Further Look at Impulse
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As mentioned in the previous
lesson
on Newton's three laws of motion, Newton's 2nd law only deals with the forces acting on a single object - that is, the
net force
. When one or more forces act on an object, the product of the sum of those forces times the duration of time over which they act equals the
change in the object's momentum
or the
impulse
received by the object.
If the forces are balanced, that is, the object is in a state of either
dynamic or static equilibrium
, then the net F equals zero and there is no change in the object's momentum - it maintains a constant velocity. However, if there is an unbalanced force, then there will be a nonzero net force and subsequently a nonzero impulse and the object will experience an
acceleration
allowing us to witness a change in the object's velocity and momentum.
Impulse
is represented by the variable
J
and has units of
N sec
.
J = (net F)t = Δp where p = mv
.
This equation can be presented graphically as
Since the impulse that an object receives equals the change in its momentum, the units for impulse and momentum must be equivalent.
impulse
Δmomentum
N sec
kg m/sec
(kg m/sec
^{2}
) sec
kg m/sec
Remember by
Newton's Third Law
that during a collision, the impulse gained by one object is exactly equal to the impulse lost by the second object - that is,
F
_{AB}
t = -F
_{BA}
t
.
In calculus terms the area under a graph is expressed as an integral. This allows us to restate the impulse as
Let's work an example using this relationship.
Refer to the following information for the next two questions.
Suppose a force, F(t) = 6t
^{2}
- 3t +1, acts on an 7-kg mass for three seconds.
What impulse will the 7-kg object receive in the first three seconds?
If the mass started from rest, what is its final velocity?
Force as the rate of change of momentum
The impulse equation
J = (net F)t = Δp where p = mv
can be rearranged to state that the applied net force applied to an object equals the rate of change of the its momentum.
That is, the net force acting on an object can be calculated as the
slope of a momentum vs time
graph. In terms of the calculus, this result equates to taking the derivative.
Notice that force must expressed as a function in terms of time, not displacement. Calculus will allow us to determine expressions for instantaneous, non-constant forces and thus is applicable to a wider range of situations.
Let's work an example using this relationship.
Refer to the following information for the next three questions.
Using the graph provided below, determine the instantaneous force acting on the 7-kg mass at each of the specified times: t = -2 seconds, t = 0 seconds, t = 3.5 seconds.
F(-2) =
F(0) =
F(3.5) =
Notice in the
force vs time
graph shown below of
F(t) = -12t
^{2}
+ 10t - 20
that not only are all of the force values negative but so is the area bounded by the graph and the t-axis.
This qualitatively agrees with the previous
momentum vs time graph
since both graphs correctly depict that the object is continuously losing momentum.
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