Question

An object of mass m is shot up an inclined slope at velocity v0. It starts at ground level. The slope
has a coefficient of kinetic friction µk, has height h, slope length d, and is inclined at an angle θ above the horizontal.

1. Find the object’s initial total energy E0
2. Find the work done by friction as the object slides from the bottom to the top of the slope
3. Find the object’s energy E at the top of the slope
4. Find the object’s final velocity v at the top of the slope

Answer 1

1)

object’s initial total energy E0 = (1/2)*m*v0^2

(2)

work done by friction as the object slides from the bottom to the top of the slope

Wf = f*d*cos180

f = frictional force = uk*m*g*costheta

Work done by friction Wf = -uk*m*g*d*costheta

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(3)

from energy conservation

E0 + Wf = E

energy at top ,E = E0 + Wf = (1/2)*m*v0^2 - uk*m*g*d*costheta

==========================

energy at top E = KE + PE

E = (1/2)*m*v^2 + m*g*h

h = d*sintheta

E = (1/2)*m*v^2 + m*g*d*sintheta

(1/2)*m*v^2 + m*g*d*sintheta = (1/2)*m*v0^2 - uk*m*g*d*costheta

v^2 = v0^2 - 2*g*d*( sintheta + uk*costheta )

v = sqrt(v0^2 - 2*g*d*( sintheta + mu_k*costheta )