11 Conservation Laws
Concepts and Principles
The Impulse-Momentum Relation
We’ve already used the impulse-momentum relation to analyze situations involving constant forces. The relation is typically applied in its component form:
Hopefully it’s not too much of a stretch to argue that for forces that vary in magnitude or direction the simple summation over a time interval (Dt) must be replaced by an integral over an infinitesimal time (dt):
Regardless of whether the forces acting on an object are constant or not, the impulse they exert on the object is precisely equal to the change in the object’s momentum.
The Work-Energy Relation
Our previous encounter with the work-energy relation resulted in:
where the forces acting on the object of interest were constant in both magnitude and direction. Again, for forces that vary, I will generalize this result to:
where the angle f is the angle between the instantaneous force acting on the object and the instantaneous displacement of the object (dr). Thus, this angle can change as the object moves along its path. Technically, this integral is termed a line integral and its evaluation can be rather complicated.
Also recall from our previous discussion of work-energy that this is not a vector equation, meaning it is not applied independently in each of the coordinate directions.
Analysis Tools
Applying the Impulse-Momentum Relation
Let’s re-examine the same situation we examined at the beginning of the previous chapter, a rocket launched directly upward with a time-dependent thrust.
A 2.0 kg toy rocket is fitted with an engine that provides a thrust roughly modeled by the function F(t) = (60 N/s) t – (15 N/s2) t2, for 0 < t < 4.0 s, and zero thereafter. The rocket is launched directly upward.
For analysis, we’ll apply the impulse-momentum relation between:
Event 1: The instant the rocket leaves the launch-pad.
Event 2: The instant the thrust drops to zero.
Remember from last chapter that the rocket does not leave the launch pad until 0.36 s after the engine is ignited.
When the engine shuts off, the rocket is traveling at 42.5 m/s upward.
We could also apply the impulse-momentum relation between:
Event 1: The instant the thrust drops to zero.
Event 2: The instant the rocket reaches its maximum height.
During this interval, the only force acting on the rocket is the force of gravity, and the impulse-momentum relation is:
Thus, the rocket reaches its highest altitude 8.34 s after launch.
It’s important to note that when a force is a function of time, it’s relatively easy to integrate the function and determine the impulse. However, it should be clear that it would not be easy to determine the work done by a force of this type. Since work is expressed as an integral of a force with respect to a displacement (dr), the force function has to be expressed in terms of position, r. In general, it’s not an easy (or sometimes possible) task to “convert” a function of time into a function of position, so work-energy is no