Gayle runs at a speed of 3.35 m/s and dives on a sled, initially at rest on the
ID: 1979465 • Letter: G
Question
Gayle runs at a speed of 3.35 m/s and dives on a sled, initially at rest on the top of a frictionless snow covered hill. After she has descended a vertical distance of 5.00 m, her brother, who is initially at rest, hops on her back and together they continue down the hill. What is their speed at the bottom of the hill if the total vertical drop is 18.0 m? Gayle's mass is 50.0 kg, the sled has a mass of 5.00 kg and her brother has a mass of 30.0 kg.
Your response is within 10% of the correct value. This may be due to roundoff error, or you could have a mistake in your calculation. Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error. m/s
Explanation / Answer
Starting at the top of the hill, find the speed of the sled plus Gayle at the instant she lands on it. From the law of conservation of momentum (perfectly inelastic collision), the velocity is: p(i) = p(f) m1v1(i) + m2v2(i) = (m1 + m2)v(f) v(f) = [m1v1(i) + m1v2(i)] / (m1 + m2) = [50.0kg)(3.35m/s) + 0] / (50.0kg + 5.00kg)= 3.04 m/s Her brother jumps on when Gayle and the sled have descended 5.00m. The velocity of Gayle plus sled at the instant her brother jumps on is found from the law of conservation of energy: E(i) = E(f) KE(i) + PE(i) = KE(f) + PE(f) 0.5mv^2(i) + mgh(i) = 0.5mv^2(f) + mgh(f) v(f) = ([v²(i) + 2g{h(i) - h(f)}])^1/2 v(f) = ([(3.04m/s)² + 2(9.80m/s²)(5.00m - 0)])^1/2 = 10.35 m/s At this instant. her brother jumps on, so we have another perfectly inelastic collision as in the first stage. Initial velocity here is 10.35m/s, m1is 55.0kg, and m2 of course is 30.0kg : v(f) = [m1v1(i) + m2v2(i)] / (m1 + m2) = [(55.0kg)(10.35m/s) + 0] / (55.0kg + 30.0kg) = 6.70 m/s Conservation of energy again gets you the final velocity at the bottom of the hill: v(f) = ([v²(i) + 2g{h(i) - h(f))^1/2 = ([(6.70m/s)² + 2(9.80m/s²)(13.0m - 0))^1/2 = 17.3m/s Don't worry about rounding errors. This is the correct approach.
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