A car's velocity as a function of time is given byvx(t)=α+βt2 v_x(t) = α + β$$t^2vx$$(t)=α+βt2, where α=3.00α = 3.00α=3.00 m/s and β=0.100β = 0.100β=0.100 m/s3. Draw vxv_x-tt and axa_x-tt graphs for the car's motion between t=0 t = 0 and t=5.00t = 5.00 s.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 15c

Chapter 2, Problem 15c

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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Understand the given velocity function v_x(t) = α + βt^2, where α = 3.00 m/s and β = 0.100 m/s^3. This function describes how the velocity of the car changes over time.

To draw the v_x-t graph, calculate the velocity at different time intervals between t = 0 and t = 5.00 s. Substitute these time values into the velocity function v_x(t) = α + βt^2 to find the corresponding velocities.

Plot the calculated velocity values on the v_x-t graph with time on the x-axis and velocity on the y-axis. This will show how the velocity changes over the given time period.

To find the acceleration function a_x(t), differentiate the velocity function v_x(t) = α + βt^2 with respect to time. The derivative of v_x(t) with respect to t gives a_x(t) = d(v_x)/dt = 2βt.

Plot the acceleration values on the a_x-t graph using the derived acceleration function a_x(t) = 2βt, with time on the x-axis and acceleration on the y-axis, for the same time interval from t = 0 to t = 5.00 s.

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Key Concepts

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

Velocity as a Function of Time

Velocity as a function of time describes how the speed of an object changes over time. In this problem, the velocity v_x(t) is given by the equation v_x(t) = α + βt^2, where α is the initial velocity and βt^2 represents the change in velocity due to acceleration. Understanding this equation is crucial for plotting the velocity-time graph.

Acceleration

Acceleration is the rate of change of velocity with respect to time. It can be found by differentiating the velocity function v_x(t) with respect to time, resulting in a_x(t) = d(v_x)/dt = 2βt. This concept is essential for drawing the acceleration-time graph, as it shows how the car's acceleration varies over the given time interval.

Graphing Motion

Graphing motion involves plotting velocity and acceleration against time to visually represent how these quantities change. For the velocity-time graph, plot v_x(t) = α + βt^2, and for the acceleration-time graph, plot a_x(t) = 2βt. These graphs help in understanding the car's motion dynamics between t = 0 and t = 5.00 s.

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Graphing Position, Velocity, and Acceleration Graphs

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An antelope moving with constant acceleration covers the distance between two points $70.0$ m apart in $6.00$ s. Its speed as it passes the second point is $15.0$ m/s. What is its acceleration?

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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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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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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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In the fastest measured tennis serve, the ball left the racquet at $73.14$ m/s. A served tennis ball is typically in contact with the racquet for $30.0$ ms and starts from rest. Assume constant acceleration. What was the ball's acceleration during this serve?

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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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A car's velocity as a function of time is given byvx(t)=α+βt2 v_x(t) = α + βt^2vx(t)=α+βt2, where α=3.00α = 3.00α=3.00 m/s and β=0.100β = 0.100β=0.100 m/s3. Draw vxv_x-tt and axa_x-tt graphs for the car's motion between t=0 t = 0 and t=5.00t = 5.00 s.

Verified video answer for a similar problem:

Key Concepts

Velocity as a Function of Time

Acceleration

Graphing Motion

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.