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 24a

Chapter 9, Problem 24a

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.

Verified step by step guidance

Understand the problem: The force on the bullet is given as a function of time, F(t) = 740 - (2.3 × 10⁵ s⁻¹) t. We are tasked with plotting a graph of F versus t for the time interval t = 0 to t = 3.0 × 10⁻³ s. To do this, we need to calculate the force at various time points within the interval and then plot the corresponding values.

Identify the key components of the equation: The force F(t) is a linear function of time t. The term 740 represents the initial force at t = 0, and the term -(2.3 × 10⁵ s⁻¹) t represents the rate at which the force decreases with time. This means the graph will be a straight line with a negative slope.

Choose time points for evaluation: Select a few time points within the interval t = 0 to t = 3.0 × 10⁻³ s (e.g., t = 0, t = 1.0 × 10⁻³ s, t = 2.0 × 10⁻³ s, and t = 3.0 × 10⁻³ s). For each time point, substitute the value of t into the equation F(t) = 740 - (2.3 × 10⁵ s⁻¹) t to calculate the corresponding force.

Calculate the force at each time point: For example, at t = 0, F(0) = 740 N. At t = 1.0 × 10⁻³ s, F(1.0 × 10⁻³) = 740 - (2.3 × 10⁵ × 1.0 × 10⁻³). Repeat this process for the other selected time points to get a set of (t, F) pairs.

Plot the graph: On a graph, place time t (in seconds) on the x-axis and force F (in newtons) on the y-axis. Plot the calculated (t, F) points and draw a straight line through them, as the relationship is linear. Label the axes and include appropriate units.

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Key Concepts

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

Force and Newton's Second Law

Force is a vector quantity that causes an object to accelerate, and it is defined by Newton's Second Law as F = ma, where m is mass and a is acceleration. In this context, the force on the bullet changes over time, which affects its acceleration and ultimately its velocity as it travels down the barrel.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the relationship between variables. In this case, plotting force (F) against time (t) allows us to see how the force exerted on the bullet changes as it moves through the barrel, providing insights into its dynamics.

Time Interval and Units

The time interval is the duration over which an event occurs, measured in seconds (s) or milliseconds (ms). Understanding the specified time interval of 0 to 3.0 ms is crucial for accurately plotting the graph and interpreting the force's behavior during the bullet's acceleration phase.

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A 22-g bullet traveling 240 m/s penetrates a 2.0-kg block of wood and emerges going 130 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

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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. Use the graph to estimate the impulse given the bullet.

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