Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 45b

Chapter 6, Problem 45b

A rifle with a barrel length of 60 cm fires a 10 g bullet with a horizontal speed of 400 m/s. The bullet strikes a block of wood and penetrates to a depth of 12 cm. How long does it take the bullet to come to rest?

Verified step by step guidance

Step 1: Identify the given values from the problem. The initial velocity of the bullet \( v_0 \) is 400 \( \text{m/s} \), the final velocity \( v_f \) is 0 \( \text{m/s} \) (since the bullet comes to rest), and the penetration depth \( d \) is 0.12 \( \text{m} \).

Step 2: Use the kinematic equation \( v_f^2 = v_0^2 + 2ad \) to solve for the acceleration \( a \). Rearrange the equation to \( a = \frac{v_f^2 - v_0^2}{2d} \). Substitute the known values into this formula to calculate \( a \).

Step 3: Once the acceleration \( a \) is determined, use the kinematic equation \( v_f = v_0 + at \) to solve for the time \( t \). Rearrange the equation to \( t = \frac{v_f - v_0}{a} \). Substitute the values of \( v_f \), \( v_0 \), and \( a \) into this formula.

Step 4: Perform the substitution and simplify the expression for \( t \). Ensure the units are consistent throughout the calculation (e.g., meters, seconds).

Step 5: Interpret the result for \( t \) as the time it takes for the bullet to come to rest after penetrating the block of wood.

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Key Concepts

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

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration. In this problem, kinematic equations can be used to determine the time it takes for the bullet to come to rest after it strikes the block of wood.

Deceleration

Deceleration refers to the rate at which an object slows down, which is a form of acceleration but in the opposite direction of the object's motion. In this scenario, the bullet experiences deceleration as it penetrates the wood, and calculating this deceleration is essential to find the time it takes to stop. The relationship between initial velocity, final velocity, and deceleration is key to solving the problem.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this case, as the bullet penetrates the wood, it loses kinetic energy, which is converted into work done against the resistance of the wood. Understanding this principle helps in calculating the depth of penetration and the time taken to come to rest by relating the initial kinetic energy of the bullet to the work done during penetration.

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