How far up the hill will it coast before starting to roll back down?

[spartan55]

Homework Statement

A car traveling at 22.0 m/s runs out of gas while traveling up a 24.0deg slope. How far up the hill will it coast before starting to roll back down?

Homework Equations

(v_final)^2 = (v_initial)^2 + 2*(a_parallel)*(x_final - x_initial)
...and some trig

The Attempt at a Solution

I know the acceleration that is important is the acceleration parallel to the plane which is found by some trig manipulation:
(a_y)sin(24.0deg) = a_parallel, where (a_y) = -9.8 m/s^2.
Thus: a_parallel = -3.986 m/s^2.
Now would I use (v_initial) = 22.0 m/s and plug this with a_parallel, x_initial = 0, and v_final = 0 m/s into the above kinematic equation to get: 60.712 m ?
For some reason this just seems like too big of an answer. Any help appreciated. Thanks.

 

[Delphi51]

That answer looks good to me!
It is interesting to check it with an energy approach. KE at the bottom equals PE at the top so
mgh = .5*mv²
Solve for h, then convert that to a distance along the slope using trigonometry.

 

[spartan55]

Wow it is a lot easier with the energy approach. We haven't talked about that yet, we're still on basic kinematics. Thanks for your help!

 

[Delphi51]

Most welcome!

 

[rwisz]

I would first clarify the assumptions and parameters of the problem. Is the car on a frictionless surface? Is there any air resistance? Are we assuming a constant acceleration or a variable one due to the changing slope? These details can greatly affect the accuracy of our calculation.

Assuming a frictionless surface and neglecting air resistance, your calculation using the kinematic equation is correct. However, as you mentioned, the answer seems quite large. This could be due to the steepness of the slope (24 degrees) and the high initial velocity (22.0 m/s). A quick sanity check would be to compare the answer to the height of the hill (assuming it is not an infinitely long hill). If the coasting distance is significantly larger than the height of the hill, it may be worth double-checking the calculations or considering the effects of air resistance.

In addition, it may be helpful to plot the motion of the car on a graph, with distance on the x-axis and time on the y-axis. This can give a visual representation of the motion and help to confirm the calculated distance.

Overall, the answer to this question will depend on the specific parameters and assumptions of the problem. It is important to carefully consider these factors and perform any necessary calculations or experiments to ensure an accurate answer.