A turtle crawls along a straight line, which we will call the xxx-axis with the positive direction to the right. The equation for the turtle's position as a function of time is x(t)=50.0x(t) = 50.0x(t)=50.0 cm + (2.002.002.00 cm/s)tt − (0.06250.06250.0625 cm/s2)$$t2t^2t2. $$Sketch graphs of xx versus tt, vxv_{x} versus tt, and axa_{x} versus tt, for the time interval t=0t = 0 to t=40t = 40 s.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 15e

Chapter 2, Problem 15e

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.

Verified step by step guidance

Understand the given position function: x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s²)t². This is a quadratic equation in terms of time, t, which describes the turtle's position along the x-axis.

To sketch the graph of x versus t, identify the type of curve. Since the equation is quadratic, the graph will be a parabola. Determine the vertex and intercepts by analyzing the coefficients and solving for x when t = 0 and when x = 0.

Find the velocity function, υx(t), by taking the first derivative of the position function x(t) with respect to time t. This will give you υx(t) = d(x(t))/dt = 2.00 cm/s − 2(0.0625 cm/s²)t.

Sketch the graph of υx versus t. The velocity function is linear, so the graph will be a straight line. Determine the slope and intercept by analyzing the coefficients of the velocity function.

Find the acceleration function, ax(t), by taking the derivative of the velocity function υx(t) with respect to time t. This will give you ax(t) = d(υx(t))/dt = −2(0.0625 cm/s²). Since the acceleration is constant, the graph of ax versus t will be a horizontal line.

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Key Concepts

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

Kinematics Equations

Kinematics involves the study of motion without considering the forces that cause it. The position equation x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s²)t² describes the turtle's motion along the x-axis. This equation is a quadratic function of time, indicating that the turtle's motion includes both constant velocity and constant acceleration components.

Velocity and Acceleration

Velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity. For the given position function, the velocity function υx(t) can be found by differentiating x(t) with respect to time, resulting in υx(t) = 2.00 cm/s - (0.125 cm/s²)t. Similarly, the acceleration ax(t) is the derivative of velocity, yielding a constant ax(t) = -0.125 cm/s².

Graphical Representation of Motion

Graphing the position, velocity, and acceleration as functions of time provides a visual understanding of the turtle's motion. The position graph x versus t is a parabola opening downwards, indicating deceleration. The velocity graph υx versus t is a straight line with a negative slope, and the acceleration graph ax versus t is a horizontal line, reflecting constant negative acceleration.

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Textbook Question

A car's velocity as a function of time is given by $v_x(t) = α + βt^2$ , where $α = 3.00$ m/s and $β = 0.100$ m/s³. Draw $v_{x}$ - $t$ and $a_{x}$ - $t$ graphs for the car's motion between $t equals 0$ and $t equals 5.00$ s.

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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?

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?

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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.

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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