Question

A sled is pushed along an ice covered lake. It has some Initial velocity before coming to rest in 15m. It took 23 seconds before the sled and rider come to rest. If the rider and sled have a combined mass of 52.5kg, what is the magnitude and direction of the stopping force?

Answer 1

If we assume the acceleration that the sled undergoes is constant throughout its motion, then we have the average velocity
`\bar v` of the sled to be

`\bar v=\frac{v_i+v_f}2=(\Delta x)/(23\,\mathrm s)`

where
`\Delta x` is the total displacement of the sled, and
`v_i` and
`v_f` are the sled's initial and final velocities, respectively. The sled eventually stops, so we take
`v_f=0` and solve for
`v_i`:

`\frac{v_i}2=(15\,\mathrm m)/(23\,\mathrm s)\implies v_i=1.3\,(\mathrm m)/(\mathrm s)`

Now, take the sled's starting position to be the origin. The sled moves in one direction, which we take to be the positive direction. Then because it's slowing down, we expect its acceleration to be in the negative direction (and hence with negative sign). In particular, the sled's position
`x` at time
`t` is

`x=x_i+v_it+\frac12at^2`

We have
`\Delta x=x-x_i=15\,\mathrm m`,
`v_i=1.3\,(\mathrm m)/(\mathrm s)`, and
`t=23\,\mathrm s`, so we can solve for acceleration
`a`:

`15\,\mathrm m=\left(1.3\,(\mathrm m)/(\mathrm s)\right)(23\,\mathrm s)+\frac12a(23\,\mathrm s)^2`

`\implies a=-0.056\,(\mathrm m)/(\mathrm s^2)`

With a mass of
`m=52.5\,\mathrm{kg}`, we find that the stopping force is

`F=ma=-2.9\,\mathrm N`

which means the stopping force has magnitude
`2.4\,\mathrm N` in the negative direction (opposite the direction of the sled's initial velocity).