AUTOMATIC 5 STARS FOR CLEAR AND CONCISE ANSWER THAT INCLUDES F.B.D (IF REQUIRED)

Question

AUTOMATIC 5 STARS FOR CLEAR AND CONCISE ANSWER THAT INCLUDES F.B.D (IF REQUIRED)

An object of mass m is free to move along the x axis on a frictionless track. In each of the cases given, it starts from rest at x = 0, t = 0. a) It moves a distance x under the action of a force F = (3 + 4x). At that point, show that the velocity of, acceleration of , and power expended on the object are given by It moves for a time t under the action of a force F (3 + 4t). At that point, show that the velocity of, acceleration of , and power expended on the object are given by

Explanation / Answer

F = m a

a = dv/dt = v dv/dx

so
m v dv/dx = 3 + 4x

m v dv = ( 3 + 4 x) dx
integrate both sides

m v^2/2 = 3x + 2 x^2

v = sqrt( 2x/m (3 + 2x))

F = ma

so a= F/m = (3 + 4x)/m

P = F v = (3+4x)*sqrt( 2x/m (3 + 2x))

b)

a = F/m = (3+4t)/m

a = dv/dt

so m dv/dt = (3 + 4t)

so m dv = 3 + 4t dt

m v = 3 t + 2 t^2
v = t/m (3 + 2t)

so P = Fv = t/m(3+2t)*(3+4t)

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