Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
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Chapter 2, Problem 116P

(a)

To determine

The velocity of the particle when its position is x=3.0m .

(a)

Expert Solution
Check Mark

Answer to Problem 116P

The velocity of particle when its position is x=3.0m is 4m/s .

Explanation of Solution

Given:

The acceleration of the particle is given by ax(x)=(2.0s2)x .

The velocity is 0m/s at x=1.0m .

Formula used:

Write the expression for the acceleration of the particle.

  a=dvdt

Here, a is the acceleration of the particle and dvdx is the rate of change velocity.

Multiply by dxdx in the above expression.

  a=dvdtdxdx=dvdxdxdt

Substitute v for dxdt and rearrange the above expression.

  a=vdvdxvdv=adx

Substitute, (2.0s2)x for a in above expression and integrate for v=0m/s to v=vm/s and x=x0 to x=x .

  v=0vvdv=x0x(2.0 s 2)xdx[v22]0v=(2.0s2)[x22]x0x

Simplify above expressinon.

  v2=(2.0s2)(x2x02)........(1)

Calculation:

Substitute 3.0m for x and 1.0m in equation (1).

  v2=(2.0s 2)[( 3.0)2( 1.0)2]v2=16v=4m/s

Conclusion:

Thus, the velocity of particle when its position is x=3.0m is 4m/s .

(b)

To determine

The time taken by the particle to travel from x=1.0m to x=3.0m .

(b)

Expert Solution
Check Mark

Answer to Problem 116P

Time taken by the particle to travel from x=1.0m to x=3.0m is 1.22s .

Explanation of Solution

Given:

The acceleration of the particle is given by ax(x)=(2.0s2)x .

The velocity is 0m/s at x=1.0m .

Formula used:

Write the expression for the velocity of the particle.

  v2=(2.0s2)(x2x02)

Further, simplify for v .

  v=(2.0s 2)(x2x02)  ........(1)

Write the expression for the instantaneous velocity of the particle.

  v=dxdt

Rearrange the above expression for dt .

  dt=dxv........(2)

Calculation:

Substitute (2.0s 2)(x2x02) for v in equation (2) and integrate for t=0s to t=ts and x=x0 to x=x .

  0tdt=x0xdx ( 2.0 s 2 )( x 2 x 0 2 )0tdt=12.0s-2x0xdx( x 2 x 0 2 )t=12.0s -2ln(x+ x 2 x 0 2 x0)

Substitute 3,0m for x and 1.0m for xo in above expression.

  t=1 2.0 s -2 ln( 3+ ( 3 ) 2 ( 1 ) 2 ( 1 ))=12ln(3+2.82)=1.22s

Conclusion:

Thus, time taken by the particle to travel from x=1.0m to x=3.0m is 1.22s .

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Chapter 2 Solutions

Physics for Scientists and Engineers

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