Final answer:
The time needed for an 850-kg car to reach 15 m/s with a power output of 40 hp without friction is calculated using the work-energy principle. When climbing a hill, further calculations include work against gravity. Newton's second law is used to elaborate on car acceleration.
Step-by-step explanation:
Time Required to Reach a Certain Speed with and without Climbing a Hill
The problem given involves an 850-kg car that must reach a speed of 15.0 m/s with a power output of 40.0 horsepower, both without friction and while climbing a 3.00-m hill. First, we would find how long it would take the car to reach this speed on a flat road by using the work-energy principle. As friction is neglected, all the power is used to increase the car's kinetic energy. The work done by the car's engine is equal to the change in kinetic energy, which can be calculated using the formula Work = 0.5 × mass × speed^2. Since power is work done per unit time, this can be rearranged to solve for time. We can calculate the time 't' by
t = (0.5 × 850 kg × (15 m/s)^2) / (40.0 hp × 746 W/hp)
For part (b), while the car is climbing a hill, the power has also to do work against gravity. The additional work is the car's weight times the height of the hill. The total work is the sum of this and the work needed to accelerate to 15 m/s. The same steps will be followed to find the time in this scenario.
To assess the acceleration of a car, consider that acceleration is derived from the force exerted divided by the mass of the car, using Newton's second law. In practice, factors such as air resistance and the car's engine efficiency can greatly affect acceleration times, thus complicating calculations.