This biology question covers important biological concepts and processes. The step-by-step explanation below helps you understand the underlying mechanisms and reasoning.
ScanSolve AI Answer
Okay, let's break down the algebra step-by-step to show how the square root appears. Step 1: Start with the conservation of energy equation. The initial potential energy of M_2 equals the final potential energy of M_1 plus the final kinetic energy of both masses. M_2 g h_2 = M_1 g h_2 + (1)/(2) M_1 v_f^2 + (1)/(2) M_2 v_f^2 Step 2: Move all terms containing potential energy to one side of the equation. Subtract M_1 g h_2 from both sides: M_2 g h_2 - M_1 g h_2 = (1)/(2) M_1 v_f^2 + (1)/(2) M_2 v_f^2 Step 3: Factor out common terms on both sides. On the left side, g h_2 is common: g h_2 (M_2 - M_1) = (1)/(2) M_1 v_f^2 + (1)/(2) M_2 v_f^2 On the right side, (1)/(2) v_f^2 is common: g h_2 (M_2 - M_1) = (1)/(2) v_f^2 (M_1 + M_2) Step 4: Isolate v_f^2. To get v_f^2 by itself, first multiply both sides by 2: 2 g h_2 (M_2 - M_1) = v_f^2 (M_1 + M_2) Now, divide both sides by (M_1 + M_2): (2 g h_2 (M_2 - M_1))/(M_1 + M_2) = v_f^2 Step 5: Solve for v_f by taking the square root of both sides. Since v_f^2 is equal to the expression, v_f is the square root of that expression: v_f = sqrt((2 g h_2 (M_2 - M_1))/(M_1 + M_2)) This is where the square root comes from. It's the final step in isolating v_f from v_f^2. Now, let's re-substitute the values: M_1 = 1.5 kg M_2 = 4.5 kg h_2 = 2 m g = 9.81 m/s^2 v_f = sqrt(2 × 9.81 m/s)^2 × 2 m × (4.5 kg - 1.5 kg)1.5 kg + 4.5 kg v_f = sqrt((2 × 9.81 × 2 × 3.0)/(6.0)) m/s v_f = sqrt((117.72)/(6.0)) m/s v_f = sqrt(19.62) m/s v_f ≈ 4.429 m/s The velocity with which M_2 hits the ground is approximately 4.43 m/s. 3 done, 2 left today. You're making progress.