Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Question: A proton moves along the axis according to the equation x = 50t + 10t² , where x is in meters and t is in seconds. Calculate (a) the average velocity of the proton during the first 3.0 s of its motion, (b) the instantaneous velocity of the proton at t = 3.0 s, and (c) the instantaneous acceleration of the proton at t = 3.0 s . (d) Graph x versus t and indicate how the answer to (a) can be obtained from the plot. (e) Indicate the answer to (b) on the graph. (f) Plot v versus t and indicate on it the answer to (c).

The average velocity of the proton during the first $3.0 s$ of its motion is $80 m/s$ .
The instantaneous velocity of the proton at $t = 3.0 s$ is $110 m/s$ .
The instantaneous acceleration of the proton at $t = 3.0 s$ is $20 {m/s}^{2}$
From graph, $x (3.0 s) = 240 m, x (0 s) = 0 m$ , Therefore, $v_{a v g} = 80 m/s$
The instantaneous velocity at $t = 3.0 s$ is $110 m/s$ .
The instantaneous acceleration from the graph is $20.57 {m/s}^{2}$ .

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TABLE OF CONTENTS :

TABLE OF CONTENTS

To understand concept

The equation for the motion of the proton is, $x = 50 t + 10 t^{2}$

Understanding the average velocity and instantaneous velocity.

Average velocity is the ratio of total displacement to the total time interval. Instantaneous velocity is the velocity of a moving object at a specific moment.

The expression for the average velocity is given as follows:

$v_{a v g} = \frac{∆ x}{∆ t}$ … (i)

Here, $∆ x$ is the displacement and $∆ t$ is the time duration.

The expression for the instantaneous velocity is given as follows:

$v = \frac{d x}{d t}$ … (ii)

The expression for the instantaneous acceleration is given as follows:

$a = \frac{d v}{d t}$ … (iii)

(a) Determination of the average velocity of the proton during the first 3.0s of its motion

Position of the proton at $t = 3.0 s$ is,

localid="1660821352042" $x (3.0 s) = 50 m/s×3.0 s+10m/s× {(3.0 s)}^{2} = 150 m+90m = 240 m$

Position of the proton at $t = 0 s$ is,

$x (0 s) = 50 m/s×0 s+10m/s× {(0)}^{2} = 0 m$

Using equation (i), the average velocity is calculated as follows:

$v_{a v g} = \frac{x (3.0 s) - x (0 s)}{t_{2} - t_{1}} = \frac{240 m-0 m}{3.0 s -0 s} = 80 m/s$

Thus, the average velocity of the proton is $80 m/s$ .

(b) Determination of the instantaneous velocity of the proton at t=3.0 s

Using equation (ii), the instantaneous velocity is,

$v = \frac{d (50 t + 10 t^{2})}{d t} v = 50 + 20 t$

The instantaneous velocity at $t = 3.0 s$ is,

role="math" localid="1650539082653" $v = 50 + 20 (3.0) = 50 + 60 = 110 m/s$

Thus, the instantaneous velocity at time $t = 3.0 s$ is $110 m/s$ .

(c) Determination of the instantaneous acceleration of the proton at t=3.0 s

Using equation (iii), the instantaneous acceleration is,

$a = \frac{d (50 + 20 t)}{d t} = 20 {m/s}^{2}$

Thus, the instantaneous acceleration at $t = 3.0 s$ is $20 {m/s}^{2}$ .

(d) Determination of average velocity from the graph.

The graph x vs t is plotted below.

From the graph,

The positions at 3.0 s and 0 s are,

$x (3.0 s) = 240 m and x (0 s) =0 m$

So, the average velocity is,

$v_{a v g} = \frac{240 m}{3.0 s} = 80 m/s$

(e) Indicate the instantaneous velocity of the proton at t=3.0 s on the plot

The instantaneous velocity at $t = 3.0 s$ is the slope of the triangle drawn at that point in the above graph and it is,

$slope= \frac{350 - 120}{4 - 1.9} \approx 110 m/s$

Thus, the instantaneous velocity at 3.0 s is 110 m/s.

(f) Plot the graph of V vs t

The graph of $v$ vs $t$ is as below,

From the graph, the instantaneous acceleration is the slope of the graph

$Slope = \frac{132 - 60}{4 - 0.5} = 20.57 {m/s}^{2}$

Thus, the instantaneous acceleration from the graph is $20.57 {m/s}^{2}$ .

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

To understand concept

Understanding the average velocity and instantaneous velocity.

(a) Determination of the average velocity of the proton during the first 3.0s of its motion

(b) Determination of the instantaneous velocity of the proton at t=3.0 s

(c) Determination of the instantaneous acceleration of the proton at t=3.0 s

(d) Determination of average velocity from the graph.

(e) Indicate the instantaneous velocity of the proton at t=3.0 s on the plot

(f) Plot the graph of V vs t

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

To understand concept

Understanding the average velocity and instantaneous velocity.

(a) Determination of the average velocity of the proton during the first 3.0s of its motion

(b) Determination of the instantaneous velocity of the proton at t=3.0 s

(c) Determination of the instantaneous acceleration of the proton at t=3.0 s

(d) Determination of average velocity from the graph.

(e) Indicate the instantaneous velocity of the proton at t=3.0 s on the plot

(f) Plot the graph of V vs t

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