My final M2 Question I need help with lol!

A particle P is projected up a line of greatest slope of a rough plane which is inclined at an angle x to the horizontal, where tan x = 3/4. The coefficient of friction between P and the plane is 1/2. The particle is projected from the point O with a speed of 10ms and come to instantaneous rest at the point A.

By using work-energy principles, or otherwise

a) Find to 3 sig fig the length OA.

b) Show that P will slide back down the plane.

c) Find to 3 sf, the speed P when it returns to O

Reply 1

dvs

a)
KE at O = 0.5m 10^2
PE at A = mg (OA sin(x))

Conservation of energy gives us:
0.5m 10^2 = mg OA sin(x)
OA = 50/(g sin(x)) = 50/(g sin(arctan(3/4))) = 8.503 m

b)
mg sin(x) - friction = mg sin(x) - 0.5 mg cos(x)
= mg (sin(x) - 0.5cos(x))
= 1.96m > 0

So the force due to gravity is greater than the friction. Therefore it must slide down.

c)
PE at A = 50m
KE at O = 0.5mv^2
External work = 1.96m

Hence work-energy gives us:
50m = 0.5mv^2 + (1.96m * 8.503)
=> v = 4.08 m/s