Gravitational Potential Energy of a vehicle

[Winegar12]

Homework Statement

An all-terrain vehicle with a 2,000 kg mass moves up a 15^o slope at a constant velocity of 6 m/s. What is the rate of change of gravitational potential energy with time?

Homework Equations

W=$$\Delta$$E/$$\Delta$$t
W=FXd
Ki+Ui=Ki+Ui (I think that's what it is)
Pe=mgh
1/2mv²=

The Attempt at a Solution

Ok I'm pretty sure those are the equations we need to use. I know what the answer is and it is in Watts. However I have tried a million things to try and get the answer but I can't seem to get it. I'm assuming that in the equation Pe=mgh, the h would change to sin$$\theta$$. To be honest, I've tried it a few different ways and can't seem to get it. I'm sure I'm overthinking it, I just need some nudge in the right direction.

 

[p21bass]

You have one of the correct equations, in PE = mgh. And yes, you have to factor in sin(theta). All of the other equations, though, aren't necessary. Since you obviously know that PE is proportional to the height - what you really need to know is how the height changes over time. Then use that information to answer the question.

 

[Winegar12]

So obviously there is another equation that I need to use obviously...any nudge on what the equation I would use? Will I be using a type of Trig equation? You said that I would need to factor in sin(theta) but I still need the height. So I won't replace the height with sin(theta), but will still use it in the equation. For example would I do, Pe =mghsin(theta) and just times them all together. The answer is a large number so I'm assuming that's what I would do...

 

[p21bass]

Almost - it won't be PE=mghsin(theta), as what you need to do is find h as a function of the distance it's traveled on the incline and sin(theta) - which is a basic trig function. Then, choose a time interval (I'd suggest 0s and 1s) and compare the two PEs.

 

[rwisz]

Thank you!

I would approach this problem by first identifying the relevant equations and variables. In this case, we are dealing with gravitational potential energy (Pe) and its rate of change with time (dPe/dt). The mass of the vehicle (m) and the height of the slope (h) are also given.

The equation for gravitational potential energy is Pe = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height above the reference point. In this case, the reference point can be chosen to be the bottom of the slope, so h would be the vertical height of the vehicle at any given time.

To calculate the rate of change of Pe with time, we can use the equation dPe/dt = mgh/t, where t is the time taken to move up the slope. However, we are given the velocity (v) of the vehicle instead of the time, so we need to find a way to relate velocity to time.

To do this, we can use the equation v = d/t, where d is the distance traveled and t is the time taken. In this case, the distance traveled up the slope (d) can be calculated using trigonometry as d = h/sinθ, where θ is the angle of the slope (15o).

Now, we have all the necessary information to calculate the rate of change of Pe with time:

dPe/dt = mgh/[d/v] = mghv/d = (2000 kg)(9.8 m/s^2)(sin15o)(6 m/s)/[h/sinθ] = 19600 W

Therefore, the rate of change of gravitational potential energy with time for this vehicle is 19600 Watts.