Set 4 Problem number 5

The velocity of an object of constant mass 16 Kg is observed to change from 8 m/s to -667 m/s in .9 seconds.

Use the Impulse-Momentum Theorem to determine the average force exerted on the object.
Verify your results using your knowledge of uniformly accelerated motion.

Solution

Since we know the initial and final velocities and the constant mass we can find the initial and final momenta of the object and therefore the change in momentum.

The initial momentum is m v0 = 16 kg * 8 m/s = 128 kg m/s.
The final momentum is m vf = 16 kg * -667 m/s = -10672 kg m/s.
The change in momentum is therefore -10672 kg m/s - 128 kg m/s = - 10800 kg m/s.

From the change in momentum and the time interval we easily calculate the average force:

We use p to stand for momentum; change in momentum is thus designated `dp and the Impulse-Momentum Theorem says that `dp = Fave `dt.
It follows from our previous calculation of momentum change that Fave = `dp / `dt.
Thus we obtain the desired average force:

Fave = `dp / `dt = =- 10800 kg m/s / ( .9 sec) = - 12000 Newtons.

Note that the initial velocity is positive, the final velocity negative.

The change in velocity can be thought of as from the initial velocity to zero, a change of - 8 m/s, then to --667 m/s, a total change of - 8 m/s - -667 m/s.

(This result can be found in the usual manner by subtracting initial velocity from final velocity, obtaining vf - v0 = --667 m/s - 8 m/s.)

This negative velocity change implies a negative force.

Here we have implicitly chosen the direction of motion as the positive direction. The negative force indicates that the force is in the negative direction--i.e., the direction opposite that of the motion. This is what we would expect, since the object is slowing down.

Generalized Solution

We know from the Impulse-Momentum Theorem that the change in momentum `dp is equal to the product Fave `dt.

If we know `dt and can find the change `dp in momentum we can then easily find Fave: Fave = `dp / `dt.

In this case we know the constant mass m of the object and its initial and final velocities v0 and vf.

We can therefore calculate the initial and final momenta: p0 = m v0 and pf = m vf.
The change in momentum is therefore `dp = pf - p0 = m vf - m v0.
It follows that the average force is

Fave = `dp / `dt = ( m vf - m v0 ) / `dt.