Answered: An acceleration vs. time graph for an…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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An acceleration vs. time graph for an object moving along a line is shown. The value of T = 8.0 s and the value of A = 11.0 m/s².

The object is moving at v = 0.8 m/s at time t=0 s from initial position x = -1.34 m. The graph shown is sinusoidal (meaning it can be written as a sine function or a cosine function) and assumed to continue for all time.

(A) Please determine the acceleration as a function of time for the object.

(B) Please determine the velocity as a function of time for the object

(C) Please determine the position as a function of time for the object

Expert Solution

Part (A)

Given

The amplitude of the graph or maximum acceleration is $A = 11 m / s^{2}$ .
The time period of the graph is $T = 8 s$ .
The initial velocity of an object is $v_{1} = 0.8 m / s$ .
The initial position of an object is $x_{1} = - 1.34 m$ .

The angular frequency is calculated as,

$\begin{array}{rcl} ω & = & \frac{2 π}{T} \\ = & \frac{2 π}{0.8 s} \\ = & 7.85 rad / s \end{array}$

The graph is shown the sine wave. The acceleration of the object is calculated as,

$a = A \sin (ω t + ϕ)$

Substitute t=0 and a=0,

$\begin{array}{rcl} 0 & = & A \sin (ω \times 0 + ϕ) \\ A \sin ϕ & = & 0 \\ \sin ϕ & = & 0 \\ ϕ & = & 0 \end{array}$

Therefore the acceleration is calculated as,

$\begin{array}{rcl} a & = & A \sin (ω t) \\ = & 11 m / s^{2} \times \sin (7.85 t) \end{array}$

Thus, the acceleration as a function of time for the object is $11 \sin (7.85 t)$ .

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