We also will find it useful to define a rate at which work is being done or energy expended in much the same way as it was useful to define a rate of change for velocity. The term used for the rate of energy expenditure is the power. It is defined as < P> = W/(t - t₀) where the instantaneous power is P = dW/dt. Hence the antiderivative of instantaneous power with respect to time is the work done. Power is measured in a number of different units, but the SI unit for power is the Watt, where 1 Watt = 1 J/s. Another commonly used unit for power is the horsepower. The relationship between horsepower and watts is

Let's do an example which makes use of this concept.

• Example 5:
  Suppose a car has an engine that can deliver 100 hp to the drive shaft. If the mass of the car plus passengers is 1800 kg and the engine maintains constant maximum power output, what is the speed of the car at the end of 2 s and at the end of 4 s? We assume that the car is on level ground and neglect friction of all types.

  Solution:
  During 2 s, the drive shaft does an amount of work equal to

  W	= (P)(t)
  Kf - K0	= (P)(t)
  ½ m(v2 - v02)	= (P)(t) ==>
  v2	= 2(P)(t)/m ==>
  v	= sqrt[2(100 hp)(746 W/hp)(2 s)/(1800 kg)]
  v	= 12.9 m/s

  assuming no friction. If we go to 4 seconds, note that the speed of the car does not double. It goes up by the sqrt[2] to 18.2 m/s.

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  Lecture Question 2

  Let's consider the car example again from a more realistic viewpoint. A compact car with a mass of 800 kg has an efficiency of about 18 percent, i.e. 0.18 of the energy in the fuel is actually delivered to moving the car wheels). If the car gets 35 mi/gal. at 60 mi/h (27 m/s), how much power is delivered to the wheels? One gallon of gasoline can produce 1.3 x 10⁸ J of energy.
  1. 11 kW
  2. 20 kW
  3. 62 kW
  4. 100 kW
  5. 2 kW

  Solution:
  We have to find the amount of energy generated by the car per unit time.

  P = (60 mi/h)(1.3×108 J/gal) (35 mi/gal.)(3600 s/h) = 62 kW

  Only 18 percent of this makes it to the wheels, so the total power delivered to the wheels is (0.18)(62 kW) = 11 kW = 15 hp. So an 83 hp engine delivers only about 15 hp to actually move the car. Vehicle mass plays a big role in determining this number, but two things are clear: car engines are remarkably inefficient and gasoline is extremely powerful stuff.

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• Example 6:
  It's clear that nature understood these principals long ago and has made remarkable use of them for efficient bioengineering. For example, consider the case of the common flea. It can jump more than 100 times its own height. By using high-speed cinephotography, researchers have determined that the flea gets from rest to a peak vertical velocity of about 1 m/s in about 1 millisecond. The maximum height attained is about 3.5 cm. To see how amazing these numbers are, you also need to know other information. A flea's mass is about 450 micrograms. About 1/5 of that mass is in the leg muscles which can develop a maximum of 60 W/kg. We can now ask some auxilliary questions.
  1. What is the average acceleration of the flea?

     The average acceleration is given by

     < a > = (v - v₀)/(Dt) = (1 m/s - 0)/(10^-3 s) = 1000 m/s²

     This is roughly 100 g's.

  2. What is the average energy loss from air resistance?

     Without air resistance, the flea would go to a maximum height of

     mgh = 1 2 mv02 Þ h = v02 2g = (1 m/s)2 2(9.8 m/s2) h = 0.051 m

     Since the maximum height reached is 0.035 m, air friction must be responsible for the loss of about (0.051 m - 0.035 m)/(0.051 m) = 30 percent on the trip up to the maximum height.

  3. Now the crucial question: can the leg muscles actually produce the energy to do the jump?

     The energy expended during the takeoff is

     P = DW Dt = (1/2)(4.5×10-7 kg)(1 m/s)2 10-3 s P = 2.25×10-4 W

     A power of 225 microwatts is enormous compared to what the muscles can produce, which is

     (60 W/kg)(4.5 x 10^-7 kg)(1/5) = 5.4 x 10^-6 W

     So how does nature do it? Through a small pad of elastic material called resilin located at the base of the large hind legs. The flea takes a few tenths of seconds to compress this material as it folds its legs in preparation for a jump. A ratchetlike structure holds the leg folded until a muscle is relaxed and the springlike structure powers the jump through stored energy. In a sense, the flea carries its own trampoline.

     Other insects like grasshoppers can be observed compressing resilin pads to power their jumps. In their case, several seconds worth of energy storage is needed to power the jumps.