The car has mass M and initial velocity v 0 which travels along the positive x-a

Question

The car has mass M and initial velocity v0 which travels along the positive x-axis with no friction.

A small rock with mass m is added to the car. What is the velocity of the car with rock? And how much did it change with initial velocity?

Next you add gravel, think of previous answer. By using previous calculation, how much did velocity of the car with gravel change (dv) when small mass is added (dm). Divide this calculation by (dt) to get the acceleration of the car. (Also, you are filling the car with gravel at a constant rate "q" kgs/s)

What is the distance traveled by the car in time t? *Could be x(t), but I'm not sure.

Assuming there is friction now, how would the previous calculated equations change? [Kinetic Friction coefficient: (Mu); Ffriction= N]

Due to the multiple calculations, I am offering more points than a normal Physics question. Thank you and good luck!

Explanation / Answer

Conservation of momentum tells us that p0 = pf .

The inital momentum is p0 = Mv0 . The momentum after the rock is added will be pf = (M + m)vf .

So Mv0 = (M + m)vf which means vf = M/(M+m)v0. The change from v0 to vf =v0 M/(M+m) is a reduction by a factor of M/(M+m) = 1/(1+(m/M)).

So if you add gravel at a constant rate of q, the total mass of gravel, m, at time t will be m=qt. If you plug this into the previous result, and use m0 instead of M, you get that the velocity as a function of time will be: v(t) = v0/(1+(qt/m0)). The distance traveled, x(t), is the integral of v(t).

x(t)= (m/q)*Ln[1+qt/m0]

where Ln is the natural logarithm function and x0 is the inital position at time t = 0.

If there is friction, instead of conservation of momentum, momentum would be changing as (dp/dt) = F = -(mu)*(m0 + qt)*g

So the momentum at time t is p(t) = (m0 + qt)*v(t)

Taking the derivative, (dp/dt) = a(t)*(m0+qt) + q * v(t) = -(mu)*(m0 + qt)*g

Simplifying:

a(t) = -(mu)*g - (q/(m0+qt))*v(t)

Or

x'' = -(q/(m0+qt))*x' -(mu)*g

This has to be solved for x(t)

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