Speed at the top of the loop with energy loss from friction

[miyayeah]

Homework Statement

If the car is going 30 m/s half way up the loop and loses energy due to friction at a rate of 2.0J per meter of track, how fast will it be going at the top?

(this is question related to the loop of roller coaster track, with radius 20m.)

Homework Equations

1. K_i + U_i = K_f + U_f
2. K_i + U_i = K_f + U_f + energy lost by friction

The Attempt at a Solution

So I found the final velocity (22.5m/s) using the first equation in the list above. I need to find Δx, so I thought of using Δx = v⋅t + ½at², but I am not sure if the acceleration here would be -9.8m/s². How would you find the acceleration in this case?

If I know the acceleration I believe I will be able to find time and Δx taken for the roller coaster to go from half way point to the top point, and then I can figure out how much J of energy is lost by friction and use equation 2 to find the final velocity.

 

[PeroK]

You are given the energy lost to friction, so you need to work out how to use that, in conjunction with your other energy calculations.

 

[miyayeah]

You are given the energy lost to friction, so you need to work out how to use that, in conjunction with your other energy calculations.

The second equation shows that, but what I did was (before using the second equation) I tried to find the curved distance of the path from the middle point to the top point because the question indicates "2.0J per meter of track". By finding the distance traveled I can multiply that by 2.0J to get the total energy lost due to friction. Also I found 22.5m/s (the final speed if no friction was present) because the number 22.5m/s can still be applied to find the distance traveled by the roller coaster... if that makes sense/if that's correct?

 

[PeroK]

The second equation shows that, but what I did was (before using the second equation) I tried to find the curved distance of the path from the middle point to the top point because the question indicates "2.0J per meter of track". By finding the distance traveled I can multiply that by 2.0J to get the total energy lost due to friction. Also I found 22.5m/s (the final speed if no friction was present) because the number 22.5m/s can still be applied to find the distance traveled by the roller coaster... if that makes sense/if that's correct?

You need to be careful talking about "displacement" when friction is involved.

Hint: How far does the car travel in one complete loop?

 

[miyayeah]

You need to be careful talking about "displacement" when friction is involved.

Hint: How far does the car travel in one complete loop?

How far the car travel would be the circumference, so :
C=2πr = 2⋅π⋅20 = 125.66 (approximately)

 

[PeroK]

How far the car travel would be the circumference, so :
C=2πr = 2⋅π⋅20 = 125.66 (approximately)

And how much energy does it lose to friction in this case?

 

[miyayeah]

And how much energy does it lose to friction in this case?

I am not sure, would the distance from the half point to the top simply be 125.66/4? If that is right, then
energy lost = 62.83 (approximately) ?

 

[PeroK]

I am not sure, would the distance from the half point to the top simply be 125.66/4? If that is right, then
energy lost = 62.83 (approximately) ?

You need to get into the habits of stating units. That must be what the question intends.