A car (m = 610.0 kg) accelerates uniformly from rest up an inclined road which rises uniformly, to a height, h = 49.0 m. Find the average power the engine must deliver to reach a speed of 24.7 m/s at the top of the hill in 20.5 s(NEGLECT frictional losses: air and rolling, ...)
m g h + (1/2) m v^2 = energy gain during trip
= 610 * 9.81 * 49 + (1/2) * 610 * 24.7^2 in Joules
divide that energy gain by 20.5 seconds to get Watts
To find the average power the engine must deliver, we can use the work-energy theorem. The work done on an object equals the change in its kinetic energy. We can calculate the work done by the engine to accelerate the car up the hill, and then divide it by the time taken to find the average power.
Let's break down the solution into steps:
Step 1: Calculate the work done to raise the car to the top of the hill.
- The work done against gravity can be calculated using the formula:
W = mgh
where W is the work done, m is the mass of the car, g is the acceleration due to gravity, and h is the height of the hill.
Plugging in the given values:
W = (610.0 kg) * (9.8 m/s^2) * (49.0 m)
Step 2: Calculate the final kinetic energy of the car at the top of the hill.
- The final kinetic energy of the car can be calculated using the formula:
KE = 0.5 * m * v^2
where KE is the kinetic energy, m is the mass of the car, and v is the final velocity of the car.
Plugging in the given values:
KE = 0.5 * (610.0 kg) * (24.7 m/s)^2
Step 3: Calculate the work done by the engine to accelerate the car.
- The work done by the engine can be calculated by subtracting the initial kinetic energy from the final kinetic energy.
W_engine = KE - 0
(Since the car starts from rest, the initial kinetic energy is 0.)
Step 4: Calculate the average power.
- Average power is defined as the work done divided by the time taken.
Power = Work / Time
Average_Power = W_engine / t
where t is the time taken to reach the top of the hill.
Plugging in the given values:
Average_Power = W_engine / 20.5 s
Finally, you can substitute the calculated values into the equations to find the average power.