What Distance Must a Rock Fall to Double Its Speed?

[swede5670]

Homework Statement

After falling 5m a rock has a velocity of v. What is the total distance the rock must fall to get a speed of 2v?

Homework Equations

GPE=mgh
Ke=1/2mv^2

The Attempt at a Solution

Should I set both equations equal to each other? I'm not even sure how to begin this one.

 

[LowlyPion]

You can do it that way. Note that v = (2*g*h)^½

If v ∝ h^½ ...

... how much then do you need to increase the height to get a doubling of speed?

 

[thomate1]

Yes, you can use the equations for gravitational potential energy (GPE) and kinetic energy (KE) to solve this problem. First, let's set up the initial situation: the rock has fallen 5m and has a velocity of v. We can use the equation for GPE to calculate the potential energy at this point: GPE = mgh, where m is the mass of the rock, g is the acceleration due to gravity (9.8 m/s^2), and h is the height (5m).

Next, we can use the equation for KE to calculate the kinetic energy of the rock: KE = 1/2mv^2, where m is the mass of the rock and v is the velocity.

Since we want to find the total distance the rock must fall to reach a speed of 2v, we can set up the final situation as follows: the rock has fallen a distance h (unknown) and has a velocity of 2v. We can use the same equations for GPE and KE, but with the new values: GPE = mgh, where m is the mass of the rock, g is the acceleration due to gravity (9.8 m/s^2), and h is the new height (unknown). And KE = 1/2mv^2, where m is the mass of the rock and v is the new velocity (2v).

To find the total distance the rock must fall, we can set the GPE in the initial situation equal to the KE in the final situation and solve for h:
mgh = 1/2mv^2
h = 0.5v^2/g

Therefore, the total distance the rock must fall is 0.5v^2/g. This means that if the rock falls a distance of 0.5v^2/g, it will have a velocity of 2v.

I hope this helps! Let me know if you have any other questions.