A 2.45 kg box, starting from rest, slides down a 6.70 m long ramp inclined at 27.5o from the horizontal. Halfway down the ramp, it hits a second box, with a mass of 1.40 kg. The two boxes stick together and slide the rest of the way down the ramp. The coefficient of kinetic friction between the ramp and the boxes is 0.150. How fast are they going when they reach the bottom?
Question
A 2.45 kg box, starting from rest, slides down a 6.70 m long ramp inclined at 27.5o from the horizontal. Halfway down the ramp, it hits a second box, with a mass of 1.40 kg. The two boxes stick together and slide the rest of the way down the ramp. The coefficient of kinetic friction between the ramp and the boxes is 0.150. How fast are they going when they reach the bottom?
Solution
To solve this problem, we'll use the principles of conservation of energy and momentum.
Step 1: Calculate the initial potential energy of the 2.45 kg box. Potential energy (PE) = mgh where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height. The height can be calculated using the sine of the angle of the ramp: h = sin(27.5) * 6.70 m = 3.01 m So, PE = 2.45 kg * 9.8 m/s^2 * 3.01 m = 72.3 J
Step 2: Calculate the final kinetic energy of the two boxes. The final kinetic energy (KE) is equal to the initial potential energy minus the work done against friction. The work done against friction (W) = friction force * distance The friction force = coefficient of friction * normal force The normal force for the first box = m1gcos(27.5), and for the two boxes = (m1+m2)gcos(27.5) So, W = 0.150 * (2.45 kg * 9.8 m/s^2 * cos(27.5) * 6.70 m / 2 + (2.45 kg + 1.40 kg) * 9.8 m/s^2 * cos(27.5) * 6.70 m / 2) = 15.8 J Therefore, KE = PE - W = 72.3 J - 15.8 J = 56.5 J
Step 3: Calculate the final velocity of the two boxes. The final kinetic energy = 1/2 * (m1+m2) * v^2 So, v = sqrt((2 * KE) / (m1+m2)) = sqrt((2 * 56.5 J) / (2.45 kg + 1.40 kg)) = 5.6 m/s
Therefore, the two boxes are moving at a speed of 5.6 m/s when they reach the bottom of the ramp.
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