A 1,600 kg car coasts down a hill starting from rest and decreases its height by 22 m over a distance of 90 m of travel. How fast will it be going at the end of the hill, if friction has no effect in slowing its motion?
To find the final speed of the car at the bottom of the hill, we can make use of the principle of conservation of energy. The total mechanical energy of the car is conserved as it moves down the hill. This means that the initial potential energy is converted into kinetic energy at the bottom of the hill.
First, let's calculate the initial potential energy (PE) of the car at the top of the hill using the equation:
PE = m * g * h
where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the hill.
PE = 1600 kg * 9.8 m/s² * 22 m
PE = 345,600 joules
Next, we can find the final kinetic energy (KE) of the car at the bottom of the hill. The kinetic energy of an object is given by the equation:
KE = 1/2 * m * v²
where v is the final velocity of the car.
Since the potential energy is converted into kinetic energy, we can equate the initial potential energy to the final kinetic energy:
PE = KE
345,600 joules = 1/2 * 1600 kg * v²
Simplifying the equation:
v² = (2 * 345,600 joules) / 1600 kg
v² = 691,200 joules / 1600 kg
v² = 432
Taking the square root of both sides:
v = √432
v ≈ 20.8 m/s
Therefore, the car will be going approximately 20.8 m/s (or about 74.9 km/h) at the end of the hill, assuming there is no friction.