This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Welcome back fiyin — missed you this week. To solve this problem, we need to use the impulse-momentum theorem, which states that the impulse applied to an object is equal to the change in its momentum. The impulse is the area under the force-time (F-t) graph. The formula for impulse is: Impulse = p = m(v_f - v_i) where p is the impulse, m is the mass, v_f is the final velocity, and v_i is the initial velocity. Step 1: Calculate the total impulse from the F-t graph. The graph can be divided into three geometric shapes: Area 1 (0 s to 3 s): A triangle. Area_1 = (1)/(2) × base × height = (1)/(2) × (3 s - 0 s) × 5 N = (1)/(2) × 3 s × 5 N = 7.5 N·s Area 2 (3 s to 5 s): A rectangle. Area_2 = width × height = (5 s - 3 s) × 5 N = 2 s × 5 N = 10 N·s Area 3 (5 s to 8 s): A triangle. Area_3 = (1)/(2) × base × height = (1)/(2) × (8 s - 5 s) × 5 N = (1)/(2) × 3 s × 5 N = 7.5 N·s The total impulse is the sum of these areas: Total Impulse = Area_1 + Area_2 + Area_3 = 7.5 N·s + 10 N·s + 7.5 N·s = 25 N·s The mass of the particle is given as m = 5 kg. Step 2: Calculate the final velocity if the particle is initially at rest. Given v_i = 0 m/s. Using the impulse-momentum theorem: Impulse = m(v_f - v_i) 25 N·s = 5 kg × (v_f - 0 m/s) 25 = 5 v_f v_f = (25)/(5) = 5 m/s Step 3: Calculate the final velocity if the particle is initially with a velocity of -4 m/s. Given v_i = -4 m/s. Using the impulse-momentum theorem: Impulse = m(v_f - v_i) 25 N·s = 5 kg × (v_f - (-4 m/s)) 25 = 5 (v_f + 4) Divide both sides by 5: 5 = v_f + 4 v_f = 5 - 4 = 1 m/s The final velocities are 5 m/s when initially at rest, and 1 m/s when initially at -4 m/s. Comparing these results with the given options, option A matches. The final answer is A. Send me the next one 📸