The drawing shows two boxes resting on frictionless ramps. One box is relatively

Question

The drawing shows two boxes resting on frictionless ramps. One box is relatively light and sits on a steep ramp. The other box is heavier and rests on a ramp that is less steep. The boxes are released from rest at A and allowed to slide down the ramps. The two boxes have masses of 10 and 38 kg. If A and B are 6.0 and 1.5 m, respectively, above the ground, determine the speed of (a) the lighter box and (b) the heavier box when each reaches B. (c) What is the ratio of the kinetic energy of the heavier box to that of the lighter box at B?

Explanation / Answer

1a) The work done will be equal to the kinetic energy at the moment of the end of the push upward. Work done = kinetic energy 2.3*10^-4 J = (1/2)*m*v^2 Solve for v. 1b) Work is calculated = Force*distance*cos(angle between them) The angle between them is zero so the cos term = 1, so it drops out. Plug in the Joules of work given and the force (average upward force in this case) and solve for the distance. 2) When the launch speed is 2v, the kinetic energy immediately after the launch is 4 times the 1st case. (because KE = (1/2*m*v^2) The launch power times the time t of the launch equals the energy delivered -- in other words the kinetic energy immediately after the launch. Let P1 be the launch power in the 1st case and P2 be the launch power in the 2nd case. Let KE1 be the energy delivered in the first case. first case energy P1*t = KE P2*t/2 = 4*KE So P2*t/2 = 4*P1*t Plug in 48.0 W for P1 and solve for P2. boxes: The height, h, of A above B is 2.4 m. Let M1 and M2 be the masses of the lighter and heavier boxes. At poimt A, the gravitational potential energies with respect to point B of the 2 boxes are GPE1 = M1*g*h GPE2 = M2*g*h At point B, the kinetic energies of the 2 boxes are KE1 = (1/2)*M1*V1^2 KE2 = (1/2)*M2*V2^2 a&b) At point B, those GPEs will be 100% converted to Kinetic Energies. So M1*g*h = (1/2)*M1*V1^2 M2*g*h = (1/2)*M2*V2^2 Solve both for the velocities.

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