Cart on a ramp: measured and theoretical energy
A 0.50 kg cart is released from rest at the top of a ramp 1.20 m above the floor. Light gates at the bottom record its speed in three trials: 4.1 m/s, 4.2 m/s, and 4.3 m/s. A second cart of 1.0 kg is released from the same height and gives speeds within 0.1 m/s of the first cart. The theoretical speed for a frictionless ramp is about 4.9 m/s. The teacher asks the students to compare the measured and theoretical energies, to explain the effect of the cart's mass, and to design a fair test of whether the ramp angle changes the final speed when the release height is kept at 1.20 m. Use g = 9.8 m/s².
1. Calculate the gravitational potential energy at release and the kinetic energy at the bottom for the 4.2 m/s trial. Account for the difference.
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The gravitational potential energy at release is mgh = 0.50 × 9.8 × 1.20 = 5.88 J. The kinetic energy at the bottom is 1/2 mv² = 0.5 × 0.50 × 4.2² = 0.25 × 17.64 = 4.41 J. The measured kinetic energy is about 1.47 J less than the potential energy lost. This missing mechanical energy is mainly converted into thermal energy by friction between the cart and the ramp, and by air resistance. Some energy may also go into rotation of the cart's wheels. The calculation assumes the cart starts from rest and that the height is measured accurately.
2. Explain why the 1.0 kg cart gives nearly the same final speed. Does friction change your answer?
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For an ideal frictionless ramp, mgh = 1/2 mv², so the mass cancels and v = √(2gh). Doubling the mass therefore does not change the theoretical final speed. The measured speeds are close for the two carts, which supports this. Friction does not completely change the conclusion because the friction force is roughly proportional to the normal force, and the normal force is proportional to mass. The deceleration caused by friction, a = F/m, is therefore roughly independent of mass for the same surfaces. The measured speed is lower than 4.9 m/s because friction removes mechanical energy, but the mass effect remains small.
3. Design a fair test of whether the ramp angle changes the final speed when the release height is kept at 1.20 m. State the variables you would control and how you would decide.
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Use the same cart, the same ramp surface, and the same light gate at the bottom. Keep the vertical release height at 1.20 m, measured with a ruler or marked stand, while changing the angle by moving the base of the ramp. Release the cart from rest each time, and repeat at least three trials for each angle. Compare the average final speed for each angle, not a single trial. Control the release position so the cart starts from rest at the same height, and keep the measurement point at the bottom the same. If the average speeds agree within the small trial-to-trial variation, the angle does not affect the final speed at fixed height; if they differ consistently, the angle has an effect. The key is to change only the angle while holding height, cart, and surface constant.