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Physics Department
Who's The Fastest - Answer
Who's The Fastest? - Answer = D
Answer: The correct answer is D. Students one and four are correct that both object have the same final kinetic energy. The block's is all linear so we have: mgh = 1/2mv^2, the mass cancels and we find a final velocity for the block of v = square root(2gh). The hoop has both linear and rotational kinetic energy, so we have mgh = 1/2mv^2 + 1/2 Iw^2, (where I = moment of inertia = mr^2, and w = angular velocity = v/r). Expressing the angular quantities in linear terms we have: mgh = 1/2mv^2 + 1/2 (mr^2)(v/r)^2. Canceling terms we get mgh = 1/2mv^2 + 1/2 mv^2 (The energy is divide equally between linear and rotational kinetic energy). Canceling the mass, combining terms, and solving for the final velocity of the hoop we get: v = square root(gh). This number is small than the velocity of the block. [sqrt(2gh) versus sqrt(gh)}]. Thus the block has the greater final linear velocity, and also reaches the bottom first. (Notice that the final velocity of the hoop does not depend on its radius.)