Set 4 Problem number 2

An object of mass 4 kg experiences a variable force F(t) (here F(t) indicates function notation, not multiplication of F by t; F(t) is the force at clock time t) for .6 seconds.

If the average force is known to be Fave = 1240 Newtons:

Use the Impulse-Momentum Theorem to find its change in velocity.
Use Newton's Second Law and your knowledge of uniformly accelerated motion to verify your result.

Solution

First we find the change in velocity using the acceleration and time interval:

The acceleration of the object will be 1240 Newtons / 4 kg = 310 m/s^2.
During the .6 seconds the velocity of the object will therefore change by

`dv = 310 m/s^2 * .6 seconds = 186 m/s.

Next we find the change in velocity using the impulse and change in momentum:

The impulse of the force is the product F * `dt = 1240 Newtons * .6 seconds = 744 Newton seconds = 744 kg m / s.
This is equal to the change in momentum, by the Impulse-Momentum Theorem.
Since the mass is unchanging the change in momentum is m `dv.
The change in velocity is therefore obtained by dividing the change in momentum by the mass:

`dv = 744 kg m/s / ( 4 kg) = 186 m/s.

Generalized Solution

If a force F acts on object of constant mass m for `dt seconds, the object will experience acceleration a = F / m for `dt seconds, resulting in velocity change

`dv = a `dt = (F / m) `dt = (F `dt) / m.

When the relationship `dv = (F `dt) / m is rearranged into the form

m `dv = F `dt

we have the Impulse-Momentum Theorem for object of constant mass.

We can use the Impulse-Momentum Theorem to find any of the quantities `dv, F, m or `dt given the values of three of these quantities.

University Physics Notes:

Using calculus we can prove that d ( mv ) = F dt, which applies to situations in which mass and/or velocity vary.

Since d ( mv ) = m dv + v dm, we have

m dv + v dm = F dt.

When mass is constant, dm = 0 so v dm disappears and we have m dv = F dt, as before.
When velocity is constant we have v dm = F dt, which tells us that to maintain a changing mass at constant velocity requires a force which depends on how quickly the mass is changing (i.e., F = v dm/dt).

Explanation in terms of Figure(s), Extension

Figure description: