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 15a

Chapter 2, Problem 15a

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.

Verified step by step guidance

Identify the given position function: x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2. This is a quadratic equation in the form of x(t) = x_0 + v_0*t + (1/2)*a*t^2, where x_0 is the initial position, v_0 is the initial velocity, and a is the acceleration.

Determine the initial position (x_0) by evaluating the position function at t = 0. This gives x_0 = 50.0 cm.

Find the initial velocity (v_0) by identifying the coefficient of the linear term in the position function. The term (2.00 cm/s)t indicates that the initial velocity v_0 is 2.00 cm/s.

Calculate the initial acceleration (a) by identifying the coefficient of the quadratic term in the position function. The term (0.0625 cm/s^2)t^2 corresponds to (1/2)*a*t^2, so solve for a: a = 2 * 0.0625 cm/s^2.

Summarize the findings: The turtle's initial position is 50.0 cm, the initial velocity is 2.00 cm/s, and the initial acceleration is 0.125 cm/s^2.

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Key Concepts

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

Kinematic Equations

Kinematic equations describe the motion of objects in terms of displacement, velocity, acceleration, and time. In this problem, the position function x(t) = 50.0 cm + (2.00 cm/s)t − (0.0625 cm/s^2)t^2 is a kinematic equation that provides the turtle's position as a function of time, allowing us to extract initial conditions and understand the motion dynamics.

Initial Conditions

Initial conditions refer to the values of position, velocity, and acceleration at the start of observation (t = 0). For the turtle, the initial position is the constant term in the equation (50.0 cm), the initial velocity is the coefficient of t (2.00 cm/s), and the initial acceleration is twice the coefficient of t^2, which is -0.125 cm/s^2, derived from the equation's structure.

Differentiation in Physics

Differentiation is a mathematical process used to find rates of change, such as velocity and acceleration, from position functions. The first derivative of the position function with respect to time gives the velocity function, and the second derivative gives the acceleration. In this problem, the coefficients of t and t^2 in the position equation directly provide the initial velocity and acceleration, respectively.

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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?

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 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 decreasing in magnitude?

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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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