High-speed motion pictures (35003500 frames/second) of a jumping, 210–μg210–μg flea yielded the data used to plot the graph in Fig. E2.542.54. (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November 19731973 Scientific American.) This flea was about 22 mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Is the acceleration of the flea ever zero? If so, when? Justify your answer.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 54a

Chapter 2, Problem 54a

High-speed motion pictures ( $3500$ frames/second) of a jumping, $210 en dash mu g$ flea yielded the data used to plot the graph in Fig. E $2.54$ . (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Is the acceleration of the flea ever zero? If so, when? Justify your answer.

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Examine the graph provided. The graph shows the velocity of the flea over time. The x-axis represents time in milliseconds (ms), and the y-axis represents velocity in meters per second (m/s).

Identify the regions of the graph where the velocity is constant. A constant velocity indicates that the acceleration is zero because acceleration is the rate of change of velocity.

Notice that after approximately 150 ms, the velocity becomes constant at around 8 m/s. This indicates that the flea's acceleration is zero during this period.

Understand that acceleration is the derivative of velocity with respect to time. When the slope of the velocity-time graph is zero, the acceleration is zero.

Conclude that the flea's acceleration is zero when the velocity is constant, which occurs after the initial acceleration phase, around 150 ms onwards.

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Key Concepts

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

Acceleration

Acceleration is defined as the rate of change of velocity over time. It can be positive, negative, or zero. In the context of the flea's jump, acceleration is crucial for understanding how the flea increases its speed during takeoff and whether it ever stops accelerating, which would indicate a moment of zero acceleration.

Speed-Time Graph

A speed-time graph visually represents an object's speed over time. The slope of the graph indicates acceleration; a flat line indicates constant speed, while a rising line indicates increasing speed. Analyzing the graph of the flea's jump helps determine when the flea's speed is constant, which corresponds to zero acceleration.

Kinematics of Jumping

Kinematics is the study of motion without considering the forces that cause it. In the case of the flea's jump, understanding the kinematics involves analyzing the initial takeoff, the peak of the jump, and the descent. This analysis helps identify the phases of motion where acceleration may be zero, particularly when the flea reaches its maximum height and momentarily stops before descending.

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Related Practice

Textbook Question

A rocket starts from rest and moves upward from the surface of the earth. For the first $10.0$ s of its motion, the vertical acceleration of the rocket is given by $a sub y equals open paren 2.80$ m/s³ $) t$ , where the $+ y$ -direction is upward. What is the height of the rocket above the surface of the earth at $t equals 10.0$ s?

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Textbook Question

High-speed motion pictures ( $3500$ frames/second) of a jumping, $210–μg$ flea yielded the data used to plot the graph in Fig. E $2.54$ . (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Find the flea's acceleration at $0.5$ ms, $1.0$ ms, and $1.5$ ms.

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Textbook Question

High-speed motion pictures ( $3500$ frames/second) of a jumping, $210–μg$ flea yielded the data used to plot the graph in Fig. E $2.54$ . (See 'The Flying Leap of the Flea' by M. Rothschild, Y. Schlein, K. Parker, C. Neville, and S. Sternberg in the November $1973$ Scientific American.) This flea was about $2$ mm long and jumped at a nearly vertical takeoff angle. Use the graph to answer this question: Find the maximum height the flea reached in the first $2.5$ ms.

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Textbook Question

A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 m/s. Would the rocket operate in outer space where there is no atmosphere? If so, how would you steer it? Could you brake it?

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Textbook Question

A rocket starts from rest and moves upward from the surface of the earth. For the first $10.0$ s of its motion, the vertical acceleration of the rocket is given by $a_{y}=(2.80$ m/s³ $)t$ , where the $+y$ -direction is upward. What is the speed of the rocket when it is $325$ m above the surface of the earth?

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Textbook Question

You throw a small rock straight up from the edge of a highway bridge that crosses a river. The rock passes you on its way down, $6.00$ s after it was thrown. What is the speed of the rock just before it reaches the water $28.0$ m below the point where the rock left your hand? Ignore air resistance.

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