# University of Wisconsin Green Bay

A child on a sled has been given a push so that she is moving up a hill. The hill is covered with icy snow and can be treated as frictionless. The child has a mass of 22.7 kg; the sled has a mass of 3.18 kg. What is the acceleration of the child as she moves up the incline? How far will she go before coming to a stop if her speed is 2 m/s when she starts at the bottom of the hill? She holds on tightly and so does not fall off the sled, and the hill makes an angle of 15.5 0 with the horizontal.

• In this problem, you are asked to relate motion (the sled moves up the hill) to force(even though forces are not explicitly stated, you know that the reason the child’s velocity changes is the gravitational force). Force and motion of a single object are always related through Newton’s Second Law, so this is a force or 2nd Law problem.

For the second part of the problem, you are asked to describe motion. The second question is a kinematics or energy problem.

• Step 1

Your FBD is not yet finished, because mg has both x- and y- components. Continue down to step 2 when you are ready to continue.

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Step 2

In the final FBD drawn here, all forces are divided into components. The contribution each force makes in the x-direction (along the incline) is shown explicitly, as is the contribution each force makes in the y-direction. The FBD is now a visual representation of ∑F=ma in each direction.

• The key equation for any problem that relates forces and motion is Newton’s Second Law. Regardless of what quantity you are asked to find, begin with the Second Law. If additional information is needed, it will become apparent as you proceed.

• The problem asks for the acceleration of the sled. In this case, you know that the sled accelerates along the incline, or in the x-direction. You do not need to work in the y-direction to solve this problem.

Scroll down when you are ready to solve for the distance the sled covers.

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Step 2

The child will travel 0.763 m downhill before coming to a stop.

• In this problem, the child and sled are on a slight hill. Therefore, gravity does not act directly into the surface. This is seen in the figure with two components of gravity—one into the hill and one along it. It is seen in the solution because the acceleration of the sled is less than that of gravity (less than the sled would have if it was in free fall.) The sign of acceleration indicates the direction of the vector and not whether or not the sled is slowing down or speeding up.

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Because the acceleration is opposite to the sled’s initial motion, the sled slows down. When it reaches a stop, the child and sled have reached their maximum distance along the hill. The negative sign on Δx indicates the final position is to the left of the starting position.