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Chapter 2: Problem 75

Consider an object traversing a distance \(L,\) part of the way at speed \(v_{1}\) and the rest of the way at speed \(v_{2} .\) Find expressions for the average speeds when the object moves at each of the two speeds (a) for half the total time and (b) for half the distance.

Short Answer

Expert verified
The average speeds for scenario (a) and (b) are \((v_{1} + v_{2})/2\) and \(2/(\(1/v_{1} + 1/v_{2}\))\), resepctively.

Step by step solution

01

Average Speed for Half the Total Time

For scenario (a), since the object moves at each speed for half the total time, the total time is divided between the two speeds. Therefore the distance covered at each speed, using the time-speed relationship \(d = vt\), is \(d_{1} = v_{1}t/2\) and \(d_{2} = v_{2}t/2\). The total distance \(L\) is then \(d_{1} + d_{2} = v_{1}t/2 + v_{2}t/2 = L\). The average speed is the total distance divided by the total time, therefore it is \((v_{1} + v_{2})/2\).
02

Average Speed for Half the Total Distance

For scenario (b), since the object moves at each speed for half the total distance, the total distance is divided between the two speeds. Therefore the time taken at each speed, using the speed-distance relationship \(t = d/v\), is \(t_{1} = L/2v_{1}\) and \(t_{2} = L/2v_{2}\). The total time \(t\) is then \(t_{1} + t_{2} = L/2v_{1} + L/2v_{2} = t\). The average speed is the total distance divided by the total time, therefore it is \(L/\(t_{1} + t_{2}\) = 2L/\(L/v_{1} + L/v_{2}\) = 2/(\(1/v_{1} + 1/v_{2}\))\).
03

Simplify the Exepression

Given that \(v_{1}\) and \(v_{2}\) are the speeds of the object in the first and second parts of the motion, the simplification of the previous expressions gives the average speed for scenario (a) as \((v_{1} + v_{2})/2\) and for scenario (b) as \(2/(\(1/v_{1} + 1/v_{2}\))\).

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