How can I solve this please help?

Question

A crate is given a push across a horizontal surface. The crate has a mass m, the push gives it an initial speed of 3.00 m/s, and the coefficient of kinetic friction between the crate and the surface is 0.145.

(a)Use energy considerations to find the distance (in m) the crate moves before it stops. ____ m

(b)Determine the stopping distance (in m) for the crate if its initial speed is doubled to 6.00 m/s. ____ m

Answer 1

Let

m be the mass of the crate,

v = 3 m/s or 6 m/s be the initial velocity,

f = 0.145 be the coefficient of friction,

g = 9.8 m/s² be gravitational acceleration,

s be the distance travelled (to be computed).

The initial kinetic energy is dissipated into friction,

that is, into the work performed by the force of friction.

The force of friction, F, is due to the crate's weight, mg:

F = m g f

That force performs work

W = F s

As stated above, that work equals the initial kinetic energy mv²/2.

Thus

m g f s = m v² / 2

s = v² / 2 g f

(a)

s = (3 m/s)² / (2 . 9.8 m/s² . 0.145) = 3.17 m

(b)

s = (6 m/s)² / (2 . 9.8 m/s² . 0.145) = 12.66 m

I think the teacher wants you to notice two things:

1) The result is independent of the mass m.

2) By doubling the speed you quadruple the distance.

My numbers do not show that exactly because of round-off.