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 68a

Chapter 2, Problem 68a

David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s² at the instant when David passes. How far does Tina drive before passing David?

Verified step by step guidance

Define the motion equations for both David and Tina. For David, since he is moving at a constant velocity, his position as a function of time is given by:

x_D = v_D \(\cdot\) t

, where

v_D = 30 \; \(\text{m/s}\)

. For Tina, since she starts from rest and accelerates uniformly, her position as a function of time is given by:

x_T = \(\frac{1}{2}\) a_T \(\cdot\) t^2

, where

a_T = 2.0 \; \(\text{m/s}\)^2

Set the positions of David and Tina equal to each other to find the time

t

when Tina passes David. This gives the equation:

v_D \(\cdot\) t = \(\frac{1}{2}\) a_T \(\cdot\) t^2

Simplify the equation to solve for

t

. Divide through by

t

(assuming

t \(\neq\) 0

v_D = \(\frac{1}{2}\) a_T \(\cdot\) t

. Rearrange to find

t

t = \(\frac{2 \cdot v_D}{a_T}\)

Substitute the known values of

v_D = 30 \; \(\text{m/s}\)

and

a_T = 2.0 \; \(\text{m/s}\)^2

into the equation for

t

to calculate the time it takes for Tina to pass David.

Use the value of

t

to find the distance Tina drives before passing David. Substitute

t

into Tina's position equation:

x_T = \(\frac{1}{2}\) a_T \(\cdot\) t^2

. This will give the distance Tina drives before passing David.

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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 scenario, understanding kinematics is essential to analyze the motion of both David and Tina, particularly how their positions change over time.

Equations of Motion

The equations of motion describe the relationship between an object's displacement, initial velocity, acceleration, and time. For Tina, who starts from rest and accelerates, the equation s = ut + 0.5at² is particularly relevant, where 's' is displacement, 'u' is initial velocity, 'a' is acceleration, and 't' is time. This equation will help determine how far Tina travels before she catches up with David.

Relative Motion

Relative motion refers to the calculation of the motion of an object as observed from a particular reference frame. In this problem, David's constant speed and Tina's accelerating motion must be analyzed relative to each other. Understanding relative motion is crucial to determine when and where Tina will pass David, as it involves comparing their positions over time.

Recommended video:

Guided course

04:27

Intro to Relative Motion (Relative Velocity)

Related Practice

Textbook Question

When a 1984 Alfa Romeo Spider sports car accelerates at the maximum possible rate, its motion during the first 20 s is extremely well modeled by the simple equation v_x² = (2P/m)t, where P = 3.6 ✕ 10⁴ watts is the car's power output, m = 1200 kg is its mass, and v_x is in m/s. That is, the square of the car's velocity increases linearly with time. Find an algebraic expression in terms of P, m, and t for the car's acceleration at time t.

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Textbook Question

David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s² at the instant when David passes. What is her speed as she passes him?

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Textbook Question

Nicole throws a ball straight up. Chad watches the ball from a window 5.0 m above the point where Nicole released it. The ball passes Chad on the way up, and it has a speed of 10 m/s as it passes him on the way back down. How fast did Nicole throw the ball?

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Textbook Question

A motorist is driving at 20 m/s when she sees that a traffic light 200 m ahead has just turned red. She knows that this light stays red for 15 s, and she wants to reach the light just as it turns green again. It takes her 1.0 s to step on the brakes and begin slowing. What is her speed as she reaches the light at the instant it turns green?

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Textbook Question

FIGURE P2.64 shows a fixed vertical disk of radius R. A thin, frictionless rod is attached to the bottom point of the disk and to a point on the edge, making angle Φ (Greek phi) with the vertical. Find an expression for the time it takes a bead to slide from the top end of the rod to the bottom.

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Textbook Question

If a Tesla Model S P100D in 'Ludicrous mode' is pushed to its limit, the first $3.0 s$ of acceleration can be modeled as

$a_{x} = {\begin{matrix} (35 {m/s}^{3}) t & 0 s \leq t \leq 0.40 s \\ 14.6 {m/s}^{2} - (1.5 {m/s}^{3}) t & 0.40 s \leq t \leq 3.0 s \end{matrix}$

What acceleration would be needed to achieve the same speed in the same time at constant acceleration? Give your answer as a multiple of $g$ .

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