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 112P

(a)

To determine

The instantaneous velocity as a function of time.

(a)

Expert Solution
Check Mark

Answer to Problem 112P

The instantaneous velocity as the function of time is given as vx=v0x+a0xt+12bt2 .

Explanation of Solution

Given:

The acceleration of the particle is ax=a0x+bt .

The position of particle is x=x0 and vx=v0x at t=0 .

Formula used:

Write expression for the acceleration of the particle.

  ax=dvxdt

Here, ax is the acceleration of the particle and dvxdt is the rate of change of velocity.

Rearrange above expression for dv .

  dv=adt  ........(1)

Calculation:

Substitute a0x+bt for ax and integrate equation (1) from v0x to vx and 0s to ts .

   v 0x v xdv=0t( a x+bt)dt[vx]v0xvx=[a0xt+12bt2]0tvxv0x=a0xt+12bt2vx=v0x+a0xt+12bt2

Conclusion:

Thus, the instantaneous velocity as the function of time is given as vx=v0x+a0xt+12bt2 .

(b)

To determine

The position of particle as function of time.

(b)

Expert Solution
Check Mark

Answer to Problem 112P

The position of particle as a function of time is given by x=xo+v0xt+a0xt22+16bt3 .

Explanation of Solution

Given:

The acceleration of the particle is ax=a0x+bt .

The position of particle is x=x0 and vx=v0x at t=0 .

Formula used:

Write expression for the instantaneous velocity of the particle.

  vx=v0x+a0xt+12bt2

Write expression for the velocity of the particle.

  vx=dxdt  ........(2)

Rearrange above expression for dx .

  dx=vxdt

Calculation:

Substitute v0x+a0xt+12bt2 for vx in equation (2) and integrate for x0 to x and 0 to t .

   x 0xdx=0t[v 0x+a 0xt+12bt2]dt[x]x0x=[v0xt+a 0xt22+16bt3]0txx0=v0xt+a0xt22+16bt3x=xo+v0xt+a0xt22+16bt3

Conclusion:

Thus, the position of particle as a function of time is given by x=xo+v0xt+a0xt22+16bt3 .

(c)

To determine

The average velocity for the time interval 0 to t .

(c)

Expert Solution
Check Mark

Answer to Problem 112P

The average velocity for the given time interval is given by vav=v0x+a0xt2+16bt2 .

Explanation of Solution

Given:

The acceleration of the particle is ax=a0x+bt .

The position of particle is x=x0 and vx=v0x at t=0 .

Formula used:

Write expression for average velocity of the particle.

  vav=1Δtt=t1t2v(t)dt  ........(3)

Calculation:

Substitute 0 for t1 , t for t2 , (t0) for Δt and xo+v0xt+a0xt22+16bt3 for v(t) .

  vav=1( t0)0t( v 0x + a 0x t+ 1 2 b t 2 )dtvav=1t[v0xt+ a 0x t 22+16bt3]0tvav=v0x+a0xt2+16bt2

Conclusion:

Thus, the average velocity for the given time interval is given by vav=v0x+a0xt2+16bt2 .

(d)

To determine

The average of initial and final velocity and compare with part (c)

(d)

Expert Solution
Check Mark

Answer to Problem 112P

The average of the initial and final velocities is given by vav=v0x+12a0xt+14bt2 . This is not equal to the average velocity for the given time interval.

Explanation of Solution

Given:

The acceleration of the particle is ax=a0x+bt .

The position of particle is x=x0 and vx=v0x at t=0 .

Formula used:

Write expression for the average instantaneous velocity of the particle.

  vav=vox+vx2  ........(4)

Calculation:

Substitute v0x+a0xt+12bt2 for vx in equation (4).

  vav=v ox+v 0x+a 0xt+12bt22vav=2v 0x+a 0xt+12bt22vav=v0x+12a0xt+14bt2

Conclusion:

Thus, the average of the initial and final velocities is given by vav=v0x+12a0xt+14bt2 . This is not equal to the average velocity for the given time interval.

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Physics for Scientists and Engineers

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Position/Velocity/Acceleration Part 1: Definitions; Author: Professor Dave explains;https://www.youtube.com/watch?v=4dCrkp8qgLU;License: Standard YouTube License, CC-BY