We already know that power is the time rate of Work done. So, at any given point in time, the power can be defined as:
dW is the work done in the time period of dt. Since, we know that W = F.d, we can rewrite the above equation as:
Here, F is the force, dx is the displacement of the object and dt is the time period of that displacement. Since, we know that v = dx/t, we can rewrite the above equation as:
F is the force and v is the instantaneous velocity of the object. Force and velocity are the vector quantities and power is the scalar product of these two vectors. Hence, we have derived the power formula in terms of force and velocity. So,
P = F v cosθ
Let us consider another example. While testing a newly developed engine, the engineers of Honda see that their 125 kg testing motorcycle propels from standstill to a top speed of 100 km per hour in 5 seconds. If the same engine is used in their racing motorcycle of mass 85 kg, how many seconds will elapse before the racing motorcycle achieves a speed of 100 km per hour?
Solution: We have: v = 100 × (1000/3600) = 27.78 m/s. We know that, a = v/t or average acceleration of the test motorcycle a = 27.78/5 = 5.56 m/s?. We also know that, F = ma. Hence Force applied by engine is F = 125 × 5.56 = 695 N. By applying the power formula, P = F v, the peak power of the engine is P = 695 × 27.78 = 19,307 W
For the Racing Motorcycle
Since, the engine is the same, its peak power will remain constant. Therefore P = 19,307W. At peak power, the speed of racing motorcycle is same, or v = 27.78 m/s. By applying the power formula, P = F v or, 19307 = F × 27.78, we get, F = 19307/27.78 = 695 N. Since, F = m a, we have 695 = 85 × a.
Therefore the acceleration of the racing motorcycle is a = 695/85 = 8.18 m/s?. Hence the time required for the racing motorcycle to reach 100 kmph is, t = v/a = 27.78/8.18 or 3.4 seconds.
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