Cart rolling into sand vs gravel using work-energy theorem

[bdh2991]

Homework Statement

A cart is set rolling across a level table, at the same speed
on every trial. If it runs into a patch of sand, the cart exerts
on the sand an average horizontal force of 6 N and travels
a distance of 6 cm through the sand as it comes to a stop.
If instead the cart runs into a patch of gravel on which
the cart exerts an average horizontal force of 9 N, how far
into the gravel will the cart roll before stopping? (a) 9 cm
(b) 6 cm (c) 4 cm (d) 3 cm (e) none of those answers

Homework Equations

w = ΔK.E.

The Attempt at a Solution

I basically found that 6N(.06) = .36 J for the sand part of it and set that answer to 9N * x = .36J and got the right answer but i don't know if i got the right answer because of the right reasons. My thinking was that if the mass and velocity of both carts are the same then the change in kinetic energy is ultimately going to be the same but I'm not to sure about this

 

[BruceW]

you've got the right idea. In both cases, the KE of the cart is the same, so the work done to stop the cart is the same in both cases.