Significance This is typical for power expenditure in strenuous exercise; in everyday units, it's somewhat more than one horsepower (1 hp = 746 W) . The power involved in moving a body can also be expressed in terms of the forces acting on it. If a force acts on a body that is displaced ? in a time dt, the power expended by the force is P dW dt dt F . d? d? -F.v, dt where V is the velocity of the body. The fact that the limits implied by the derivatives exist, for the motion of a real body, justifies the rearrangement of the infinitesimals. Automotive Power Driving Uphill How much power must an automobile engine expend to move a 1200-kg car up a 15% grade at 90 km/h (see figure below)? Assume that 25% of this power is dissipated overcoming air resistance and friction. v= 90 km/h m = 1200 kg 15% grade We want to calculate the power needed to move a car up a hill at constant speed. Strategy At constant velocity, there is no change in kinetic energy, so the net work done to move the car is zero. Therefore the power supplied by the engine to move the car equals the power expended against gravity and air resistance. By assumption, 75% of the power is supplied against gravity, which equals mg · v = mgu sin 0, where 0 is the angle of the incline. A 15% grade means tan 0 = 0.15. This reasoning allows us to solve for the power required. Solution Carrying out the suggested steps, we find 0.75 P = mgu sin (tan-1 0.15), or P = (1200 x 9.8 N) (90 m/3.6 s) sin (8.53") = 58 kW, 0.75 or about 78 hp. (You should supply the steps used to convert units.)
Significance This is a reasonable amount of power for the engine of a small to mid-size car to supply (1 hp = 0.746 kW). Note that this is only the power expended to move the car. Much of the engine's power goes elsewhere, for example, into waste heat. That's why cars need radiators. Any remaining power could be used for acceleration, or to operate the car's accessories. Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction. R6_10 1/1 point (graded) A barbell is very heavy object used in weightlifting. It consists of a horizontal bar and heavy disks attached to each end. Estimate the power expended by a weightlifter raising a 150-kg barbell 2 m in 3 s. V 980 980 Key Equations Work done by a force over an infinitesimal displacement dw =7 .at - F|a |cose Work done by a force acting along a path from A to B WAS = DALLAS Work done by a constant force of kinetic friction Wa =- fi llas! Work done going from A to B by Earth's gravity, near its surface Www.AB = - mg (ys - VA) Work done going from A to B by one-dimensional spring force Wspring AB