Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 15b

Chapter 2, Problem 15b

A turtle crawls along a straight line, which we will call the $x$ -axis with the positive direction to the right. The equation for the turtle's position as a function of time is $x(t) = 50.0$ cm + ( $2.00$ cm/s) $t$ − ( $0.0625$ cm/s²) $t^2$ . At what time $t$ is the velocity of the turtle zero?

Verified step by step guidance

Start by understanding the given position function: x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s²)t². This function describes the turtle's position on the x-axis as a function of time.

To find the time when the velocity is zero, first determine the expression for velocity. Velocity is the derivative of the position function with respect to time. Differentiate x(t) to get v(t).

The derivative of x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s²)t² is v(t) = d(x)/dt = 2.00 cm/s − 2(0.0625 cm/s²)t.

Set the velocity function v(t) equal to zero to find the time when the velocity is zero: 0 = 2.00 cm/s − 2(0.0625 cm/s²)t.

Solve the equation 0 = 2.00 cm/s − 2(0.0625 cm/s²)t for t. This will give you the time at which the turtle's velocity is zero.

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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 physics that deals with the motion of objects without considering the forces that cause the motion. It involves parameters such as position, velocity, and acceleration. In this problem, the turtle's position is given as a function of time, which allows us to derive its velocity and acceleration.

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It is the first derivative of the position function with respect to time. For the turtle, the velocity function can be found by differentiating the given position function x(t) with respect to time t.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In physics, it is used to find rates of change, such as velocity and acceleration. To find when the turtle's velocity is zero, we take the derivative of the position function to get the velocity function and solve for when this derivative equals zero.

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Related Practice

Textbook Question

The table shows test data for the Bugatti Veyron Super Sport, the fastest street car made. The car is moving in a straight line (the $x$ -axis).

(a) Sketch a $v_{x}$ - $t$ graph of this car's velocity (in mi/h) as a function of time. Is its acceleration constant?

(b) Calculate the car's average acceleration (in m/s2) between (i) $0$ and $2.1$ s; (ii) $2.1$ s and $20.0$ s; (iii) $20.0$ s and $53$ s. Are these results consistent with your graph in part (a)? (Before you decide to buy this car, it might be helpful to know that only $300$ will be built, it runs out of gas in $12$ minutes at top speed, and it costs more than $dollar sign 1.5$ million!)

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Textbook Question

An astronaut has left the International Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a $10$ -s interval. What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the positive direction is to the right.

(a) At the beginning of the interval, the astronaut is moving toward the right along the $x$ -axis at $15.0$ m/s, and at the end of the interval she is moving toward the right at $5.0$ m/s.

(b) At the beginning she is moving toward the left at $5.0$ m/s, and at the end she is moving toward the left at $15.0$ m/s.

(c) At the beginning she is moving toward the right at $15.0$ m/s, and at the end she is moving toward the left at $15.0$ m/s.

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Textbook Question

A race car starts from rest and travels east along a straight and level track. For the first $5.0$ s of the car's motion, the eastward component of the car's velocity is given by $v_{x} (t) =$ ( $0.860$ m/s³)t². What is the acceleration of the car when $v sub x equals 12.0$ m/s?

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Textbook Question

A turtle crawls along a straight line, which we will call the $x$ -axis with the positive direction to the right. The equation for the turtle's position as a function of time is $x(t) = 50.0$ cm + ( $2.00$ cm/s) $t$ − ( $0.0625$ cm/s²) $t^2$ . Sketch graphs of $x$ versus $t$ , $v_{x}$ versus $t$ , and $a_{x}$ versus $t$ , for the time interval $t equals 0$ to $t equals 40$ s.

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Textbook Question

A car's velocity as a function of time is given by $v_{x} (t) = α + β t^{2}$ , where $alpha equals 3.00$ m/s and $beta equals 0.100$ m/s³. Calculate the average acceleration for the time interval $t equals 0$ to $t equals 5.00$ s.

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Textbook Question

A turtle crawls along a straight line, which we will call the $x$ -axis with the positive direction to the right. The equation for the turtle's position as a function of time is $x of t equals 50.0$ cm + ( $2.00$ cm/s) $t$ − ( $0.0625$ cm/s²) $t^{2}$ . Find the turtle's initial velocity, initial position, and initial acceleration.

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