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Ch 11: Impulse and Momentum
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 11: Impulse and MomentumProblem 10b
Chapter 11, Problem 10b

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. At what time does the rocket reach its maximum speed? What is the maximum speed?

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Step 1: Analyze the graph provided. The thrust force (F) is plotted as a function of time (t). The graph shows that the thrust force is constant at 800 N from t = 0 to t = 10 s, then decreases linearly to 0 N from t = 10 s to t = 25 s, and remains at 0 N from t = 25 s to t = 30 s.
Step 2: Recall Newton's second law of motion, which states that the net force acting on an object is equal to the rate of change of its momentum. Since the mass of the rocket is constant, the acceleration (a) can be calculated using the formula: a=Fm, where F is the thrust force and m is the mass of the rocket.
Step 3: To find the velocity of the rocket as a function of time, integrate the acceleration over time. The velocity change (Δv) can be calculated using the formula: Δv=adt. Break the integration into two intervals: (1) t = 0 to t = 10 s, where the force is constant, and (2) t = 10 s to t = 25 s, where the force decreases linearly.
Step 4: For the interval t = 0 to t = 10 s, calculate the velocity change using the constant acceleration formula: Δv=at. For the interval t = 10 s to t = 25 s, use the linear force equation to express acceleration as a function of time and integrate to find the velocity change.
Step 5: The rocket reaches its maximum speed when the thrust force becomes zero (at t = 25 s). Add the initial velocity (75 m/s) to the total velocity change calculated from both intervals to determine the maximum speed. The time at which the maximum speed occurs is t = 25 s.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thrust Force

Thrust force is the force exerted by a rocket's engines to propel it forward. It is generated by expelling mass in the opposite direction, according to Newton's third law of motion. In the context of the question, the thrust force varies over time, as shown in the provided graph, which affects the rocket's acceleration and ultimately its maximum speed.
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Acceleration

Acceleration is the rate of change of velocity of an object. In this scenario, the rocket's acceleration can be determined by dividing the thrust force by the rocket's mass, as per Newton's second law (F = ma). The varying thrust force over time indicates that the acceleration will also change, influencing how quickly the rocket can reach its maximum speed.
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Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. In this problem, kinematic equations can be used to calculate the rocket's final velocity after a certain time, given its initial velocity and the acceleration derived from the thrust force. Understanding kinematics is essential for determining the maximum speed of the rocket after the thrust phase.
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Related Practice
Textbook Question

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FIGURE EX11.6 is an incomplete momentum bar chart for a collision that lasts 10 ms. What are the magnitude and direction of the average collision force exerted on the object?

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A force in the +x-direction increases linearly from 0 N to 9000 N in 5.0 s, then suddenly ends. What impulse does this force provide?

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A 2.0 kg object is moving to the right with a speed of 1.0 m/s when it experiences the force shown in FIGURE EX11.8. What are the object's speed and direction after the force ends?

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