Ch 02: Kinematics in One Dimension

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 02: Kinematics in One Dimension

Problem 60b

Chapter 2, Problem 60b

You are 9.0 m from the door of your bus, behind the bus, when it pulls away with an acceleration of 1.0 m/s². You instantly start running toward the still-open door at 4.5 m/s. What is the maximum time you can wait before starting to run and still catch the bus?

Verified step by step guidance

Step 1: Define the motion equations for both the bus and the runner. The bus starts from rest and accelerates uniformly, so its position as a function of time is given by:

x_b = (1/2) a t^2

, where

a = 1.0 \, \(\text{m/s}\)^2

. The runner moves at a constant velocity, so their position as a function of time is:

x_r = v_r (t - t_{delay})

, where

v_r = 4.5 \, \(\text{m/s}\)

and

t_{delay}

is the time the runner waits before starting to run.

Step 2: Set up the condition for the runner catching the bus. The runner catches the bus when their position equals the bus's position, i.e.,

x_r = x_b

. Substituting the motion equations, we get:

v_r (t - t_{delay}) = (1/2) a t^2

Step 3: Solve for

t_{delay}

. Rearrange the equation to isolate

t_{delay}

t_{delay} = t - \(\frac{(1/2) a t^2}{v_r}\)

. This represents the maximum time the runner can wait before starting to run.

Step 4: Determine the range of

t

values that satisfy the physical constraints. The runner starts 9.0 m behind the bus, so the initial condition is

x_r = 9.0 \, \(\text{m}\)

when

t = 0

. Use this to ensure the solution is consistent with the problem setup.

Step 5: Substitute the given values (

a = 1.0 \, \(\text{m/s}\)^2

v_r = 4.5 \, \(\text{m/s}\)

, and

x_r = 9.0 \, \(\text{m}\)

) into the equations to calculate the maximum

t_{delay}

. Ensure the units are consistent throughout the calculation.

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Key Concepts

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

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from a particular reference point. In this scenario, the bus and the person running towards it are in motion relative to each other. Understanding how to analyze their speeds and accelerations relative to one another is crucial for determining the time it takes for the person to reach the bus.

Kinematics Equations

Kinematics equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations will help calculate the distance the bus travels while the person runs, allowing us to find the maximum waiting time before the person starts running.

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. In this case, the bus accelerates at 1.0 m/s², which means its speed increases over time. Understanding how acceleration affects the distance covered by the bus during the time the person waits is essential for solving the problem of catching the bus.

Recommended video:

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Intro to Acceleration

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