A particle of mass starts up a frictionless slope, with a velocity, V, along the slope....?

Question

What is the maximum distance, L, the particle will move up the slope? (Θ, V, m, g are known)

Thanks! (teacher is horrible and cannot teach whatsoever)

Answer 1

The law of conservation of energy applies to this problem, i.e., the kinetic energy of the particle at the bottom of the slope is equal to its potential energy when it has travelled the distance "L" along the slope.

This being said,

Kinetic energy = Potential energy

KE = kinetic energy = (1/2)mV^2

PE = potential energy = mgh

where

m = mass of the particle

V = velocity of the particle at the bottom of the slope

g = acceleration due to gravity

h = vertical height that the particle will reach

Sine KE = PE,

(1/2)mV^2 = mgh

since "m" appears on both sides of the equation, it will simply cancel out hence

(1/2)V^2 = gh

NOTE that h = LsinΘ. therefore the above equation becomes

(1/2)V^2 = gLsinΘ

and solving for "L",

L = V^2/(2gsinΘ)