A ball of mass m is tied to a cart of mass 4m by a string that passes over a pulley. The cart is on wheels, which allows us to neglect friction...

Hello!

Denote the downward speed of a ball as A cart and a point on an outer edge of a pulley have the same speed. I'll use a downward y-axis, starting from the initial position of a ball.

It is known that the kinetic energy of a rotating disk is and for linearly moving ball and cart it is and The potential energy with respect to the y-axis I use is and the total energy of the system is conserved. This way we obtain an equation

Recall and and obtain

or

This is a differential equation because It is simple to solve it and the solution is So the starting acceleration is and the answer for A is f.

For B use Newton's Second law, Net force here is equal to where is the tension force, so the answer is g.

For C is the same idea but applied to a cart, the answer is the answer is d.

D (about kinetic energy). We saw that the kinetic energies have the ratio 1/2:2:1/2, i.e. 1:4:1 or 1/6:2/3:1/6, so the answer is c.

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