Ch. 09 - Linear Momentum

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 09 - Linear Momentum

Problem 24b

Chapter 9, Problem 24b

The force on a bullet along the barrel of a firearm is given by the formula F = [740 ― (2.3 x 10⁵ s⁻¹ ) t] N over the time interval t = 0 to t = 3.0 x 10⁻³ s. Plot a graph of F versus t for t = 0 to t = 3.0 ms. Use the graph to estimate the impulse given the bullet.

Verified step by step guidance

Step 1: Understand the problem. The force on the bullet is given as a function of time: F(t) = 740 - (2.3 × 10⁵ s⁻¹) t. The task is to plot F versus t for the time interval t = 0 to t = 3.0 × 10⁻³ s and then estimate the impulse using the graph.

Step 2: Create a table of values for F(t) at different time points within the interval t = 0 to t = 3.0 × 10⁻³ s. Choose several time points (e.g., t = 0, t = 1.0 × 10⁻³ s, t = 2.0 × 10⁻³ s, t = 3.0 × 10⁻³ s) and calculate F(t) for each using the formula F(t) = 740 - (2.3 × 10⁵ s⁻¹) t.

Step 3: Plot the graph of F versus t. Use the calculated values of F(t) from Step 2 to create a graph. The x-axis represents time (t) in seconds, and the y-axis represents force (F) in newtons. Connect the points to form a straight line, as the equation is linear in t.

Step 4: Estimate the impulse from the graph. Impulse is the area under the F versus t curve. Since the graph is a straight line, the area under the curve forms a trapezoid. Use the formula for the area of a trapezoid: A = 0.5 × (base1 + base2) × height, where base1 and base2 are the force values at t = 0 and t = 3.0 × 10⁻³ s, and height is the time interval (3.0 × 10⁻³ s).

Step 5: Verify the units of impulse. Impulse is the product of force and time, so its unit is N·s (newton-seconds). Ensure that the calculated area under the curve has consistent units to represent impulse correctly.

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Key Concepts

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

Force and Time Relationship

The force acting on an object can vary over time, as described by the equation F = [740 - (2.3 x 10⁵ s⁻¹) t]. This formula indicates that the force decreases linearly as time increases, which is crucial for understanding how the bullet is accelerated within the barrel of the firearm. Analyzing this relationship helps in visualizing how the force changes and affects the bullet's motion.

Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can be calculated as the area under the force versus time graph. In this context, estimating the impulse given to the bullet involves integrating the force over the specified time interval, which provides insight into the bullet's final velocity and momentum.

Graphing Techniques

Graphing techniques are essential for visualizing mathematical relationships, such as the force versus time graph in this problem. By plotting the force values against time, one can easily identify trends, calculate areas, and derive important physical quantities like impulse. Understanding how to accurately represent and interpret graphs is vital for solving problems in physics.

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Textbook Question

A 144-g baseball moving 28.0 m/s strikes a stationary 4.85-kg brick resting on small rollers so it moves without significant friction. After hitting the brick, the baseball bounces straight back, and the brick moves forward at 1.10 m/s. What is the baseball’s speed after the collision?

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Textbook Question

A 195-kg projectile, fired with a speed of 116 m/s at a 60.0° angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine the velocity of the third fragment immediately after the explosion.

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Textbook Question

A 144-g baseball moving 28.0 m/s strikes a stationary 4.85-kg brick resting on small rollers so it moves without significant friction. After hitting the brick, the baseball bounces straight back, and the brick moves forward at 1.10 m/s. Find the total kinetic energy before and after the collision.

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Textbook Question

The force on a bullet along the barrel of a firearm is given by the formula F = [740 ― (2.3 x 10⁵ s⁻¹ ) t] N over the time interval t = 0 to t = 3.0 x 10⁻³ s. Plot a graph of F versus t for t = 0 to t = 3.0 ms.

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Textbook Question

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Textbook Question

Croquet ball A moving at 4.3 m/s makes a head-on collision with ball B of equal mass initially at rest. Immediately after the collision, ball B moves forward at 3.0 m/s. What fraction of the initial kinetic energy is lost in the collision?

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