Far in space, where gravity is negligible, a 425 kg rocket traveling at 75 m/s in the +x-direction fires its engines. FIGURE EX11.10 shows the thrust force as a function of time. The mass lost by the rocket during these 30 s is negligible. (b) At what time does the rocket reach its maximum speed? What is the maximum speed?
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Step 1: Identify the initial conditions of the problem. The rocket has a mass of 425 kg and an initial velocity of 75 m/s in the +x-direction.
Step 2: Analyze the thrust force vs. time graph. The thrust force is 800 N from 0 to 10 seconds, then decreases linearly to 0 N from 10 to 25 seconds.
Step 3: Calculate the impulse provided by the thrust force. Impulse is the area under the force-time graph. For the first 10 seconds, the area is a rectangle (800 N * 10 s). For the next 15 seconds, the area is a triangle (0.5 * 800 N * 15 s).
Step 4: Use the impulse-momentum theorem, which states that the impulse is equal to the change in momentum. Calculate the total impulse and then use it to find the change in velocity (Δv = Impulse / mass).
Step 5: Determine the final velocity by adding the change in velocity to the initial velocity. The time at which the rocket reaches its maximum speed is when the thrust force becomes zero (at 25 seconds).