A block is given a very brief push up a 20.0° frictionless incline to give it an initial speed of 12.0 m/s. (a) how far along the surface of the plane does the block slide before coming to rest?

First we calculate the time it took before coming to rest. We use the vertical component of velocity in this case. The formula is:

vy = v0 sinθ - g t

where, vy is final velocity at the peak = 0, v0 sinθ = initial vertical velocity, g is acceleration due to gravity, t is time

therefore:

0 = 12 sin20 - 9.8 t

t = 0.4188 s

Then calculate the distance using the horizontal component:

d = v0 cosθ t

d = 12 cos20 * 0.4188

d = 4.72 m