University of Wisconsin Green Bay

While driving in the mountains, you notice that when the freeway goes steeply down hill, there are emergency exits every few miles. These emergency exits are straight dirt ramps which leave the freeway and are sloped uphill. They are designed to stop trucks and cars that lose their brakes on the downhill stretches of the freeway even if the road is covered in ice. You are curious, so you stop at the next emergency road. You estimate that the road rises at an angle of 100 from the horizontal and is about 100 yards (300 ft) long. What is the maximum speed of a truck that you are sure will be stopped by this road, even if the frictional force of the road surface is negligible?
This problem was used with permission of Dr. Ken Heller of the University of Minnesota Physics Education Research Group.

  • In this problem, you are asked to relate motion (the truck comes to a stop) to force (gravity). Force and motion of a single object are always related through Newton's Second Law, so this is a force or 2nd Law problem even though you are asked about velocity! You should always approach a problem first by thinking about the key interactions described, regardless of what quantity you are asked to find. Going from acceleration to velocity will be a second step in this problem.

  • Step 1

    reference frames

    Your free body diagram is not yet complete because mg has both x- and y- components. Scroll down when you are ready to continue.


    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 ΣF=ma in each direction.

  • The key equation for any problem that relates forces and motion is Newton's Second Law. The left side of the equation takes into account the forces that act on the object, and the right side shows the effect of those forces. 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 even when that additional information is the quantity you are asked to find.

  • If there is no friction acting on the truck, it will slow with an acceleration of 5.6 ft/s2 because of the gravitational force. The problem asked you to find velocity. Scroll down to use this acceleration as you relate velocity and stopping distance.


    Step 2

    Now that you know the acceleration of the truck, you can use the kinematic (descriptive) equations to related velocity and stopping distance. As for the first step, I made the math easier by choosing one axis along the direction of the truck's acceleration. A useful picture for kinematic problems includes a picture of the motion and values of any descriptive variables that you know.

    As long as the truck enters the ramp traveling less than 58 ft/s, it will be able to stop by the end of the ramp even without friction.

  • As long as a truck, or any vehicle, travels less than 58 ft/s or 39 mph, it can stop without friction on the upward incline. Although you may not have the experience or intuition to know if this speed makes sense given the length and slope of the incline, you can recognize that it is a reasonable design speed for the situation.

    If you did not initially recognize that this is a two part problem (in which you solve Newton's 2nd Law for acceleration and then use that acceleration to describe the motion), make sure to look at the overall solution now to see how the two parts fit together.