Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 13b

Chapter 2, Problem 13b

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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Textbook Question

A physics professor leaves her house and walks along the sidewalk toward campus. After $5$ min, it starts to rain, and she returns home. Her distance from her house as a function of time is shown in Fig. E $2.10$ . At which of the labeled points is her velocity constant and negative?

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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?

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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