8.2: Analysing problems with impulse
Although the principle of impulse and momentum is generally valid, it is particularly useful for analysing the dynamics under these conditions:
- The time interval \(\Delta t=t_{2}-t_{1}\) during which a force acts on a point mass is very short.
- The force \(\overrightarrow{\boldsymbol{F}}_{i j}\) acting on the point mass \(i\) is very high compared to all other forces.
When these two conditions hold, e.g. during the short instance when a ball is hit by a baseball bat, we can apply 2 assumptions:
- The distance the point mass travels during the short time \(\Delta t\) is approximately zero.
- The impulse of all other forces can be neglected because they are much smaller than the force \(\overrightarrow{\boldsymbol{F}}_{i j}\).
As a consequence of point 1 , we retain information on the position of the mass without having to determine (integrate) the motion during the application of the impulse because the mass does not move. As a consequence of point 2 , we can determine the impulse by integrating only force \(\overrightarrow{\boldsymbol{F}}_{i j}\) during the time interval \(\Delta t\) and neglect all other forces.
Often segmented motion (see Sec. 5.10) is used to analyse problems with impulse and momentum. The motion in the segments before and after the impulse is analysed with the methods from the previous chapters, and only the short segment where the large force is exerted is analysed with impulse and momentum.