Physics For Scientists And Engineers: Foundations And Connections, Extended Version With Modern Physics
Physics For Scientists And Engineers: Foundations And Connections, Extended Version With Modern Physics
1st Edition
ISBN: 9781305259836
Author: Debora M. Katz
Publisher: Cengage Learning
Question
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Chapter 2, Problem 80PQ

(a)

To determine

The depth of the chasm.

(a)

Expert Solution
Check Mark

Answer to Problem 80PQ

The depth of the chasm is 46.1 m .

Explanation of Solution

Assume that the pebble falls a distance d into the chasm in a time interval Δt1 and the sound of impact travels upward the same distance in a time interval Δt2 before the rock climber hears it.

Write the equation for the total time interval.

Δt=Δt1+Δt2

Here, Δt is the total time interval, Δt1 is the time taken for the pebble to fall into the bottom of the chasm and Δt2 is the time taken for the sound impact of the pebble to reach the rock climber.

Rewrite the above equation for Δt1 .

Δt1=ΔtΔt2 (I)

Write the Newton’s equation of motion for the fall of the pebble into the chasm.

  d=vi(Δt1)+12a(Δt1)2

Here, d is the distance travelled by the pebble before reaching the bottom of the chasm, vi is the initial speed of the pebble and a is the acceleration of the pebble.

The initial speed of the pebble is zero. The pebble is under free fall during the time interval Δt1 so that its acceleration is equal to the acceleration due to gravity.

Replace vi by 0 and a by g in the above equation to get the final expression for d .

d=0+12g(Δt1)2=12g(Δt1)2 (II)

The sound from the impact travels at constant speed during the time interval Δt2 and covers the distance d .

Write the expression for distance covered during the time interval Δt2 .

d=vsΔt2 (III)

Here, vs is the speed of sound.

Equate equations (II) and (III).

  12g(Δt1)2=vsΔt2

Put equation (I) in the above equation.

  12g(ΔtΔt2)2=vsΔt2(Δt)22(Δt)(Δt2)+(Δt2)2=2vsΔt2g(Δt2)22(Δt+vsg)Δt2+(Δt)2=0

The above equation is quadratic in Δt2 .

Write the expression for the solution of the above equation.

Δt2=2(Δt+vsg)±(2(Δt+vsg))24(Δt)22 (IV)

Conclusion:

It is given that the total time interval is 3.20 s , the speed of sound is 343 m/s and the value of acceleration due to gravity is 9.81 m/s2 .

Substitute 3.20 s for Δt, 343 m/s for vs and 9.81 m/s2. for g in equation (IV) to find Δt2 .

  Δt2=2(3.20 s+343 m/s9.81 m/s2)±(2(3.20 s+343 m/s9.81 m/s2))24(3.20 s)22=0.134 s

Substitute 343 m/s for vs and 0.134 s for Δt2 in equation (III) to find d .

  d=(343 m/s)(0.134 s)=46.1 m

Therefore, the depth of the chasm is 46.1 m .

(b)

To determine

The percentage of error that would result from assuming the speed of sound to be infinite.

(b)

Expert Solution
Check Mark

Answer to Problem 80PQ

The percentage of error that would result from assuming the speed of sound to be infinite is 8.89 % .

Explanation of Solution

If the travel time for the sound of impact is ignored, it would mean that equation (II) has to be used to determine the depth.

Write the equation for the percentage of error.

% of error=|true valueobtained valuetrue value|×100% (V)

Conclusion:

Substitute 9.81 m/s2. for g and 3.20 s for Δt in equation (II) to find d .

  d=12(9.81 m/s2)(3.20 s)2=50.2 m

Substitute 46.1 m for true value and 50.2 m for obtained value in equation (V) to find the percentage of error.

  % of error=|46.1 m50.2 m46.1 m|×100%=8.89 %

Therefore, the percentage of error that would result from assuming the speed of sound to be infinite is 8.89 % .

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Chapter 2 Solutions

Physics For Scientists And Engineers: Foundations And Connections, Extended Version With Modern Physics

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