A car is driven at 80km/h on a horizontal road. If the car is now driven up 10m high rise, what speed would it have at the top of the rise? ( Assume thet the wheels of the car roll at all times so thatit dosen't experience sliding friction ; exclude any other dissipative forces )
Neglect air and tire friction and assume conservation of mechanical energy. Convert V1 (the initial velocity) from 80 km/h to meters/second. You have to assume that the car is "in neutral" so that the engine is not continuing to provide power. They should have told you that. Most cars continue to provide some power even when the foot is off the gas, so that the engine does not die.
(1/2)M (V1^2 - V2^2) = M g H
Solve for the final velocity V2. Note that M cancels out. H = 10 meters
To find the speed of the car at the top of the rise, we can use the principle of conservation of energy.
The total mechanical energy of the car at any point along its path is the sum of its kinetic energy and potential energy.
At the bottom of the rise, the car has only kinetic energy, given by the equation:
KE1 = (1/2) * m * v1^2
Where KE1 is the initial kinetic energy, m is the mass of the car, and v1 is the initial velocity (80 km/h).
When the car reaches the top of the rise, it has both kinetic energy and potential energy. The potential energy can be calculated using the equation:
PE = m * g * h
Where PE is the potential energy, m is the mass of the car, g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the rise (10 m).
Since no dissipative forces are acting on the car, the total mechanical energy is conserved. Therefore, the total mechanical energy at the bottom of the rise is equal to the total mechanical energy at the top of the rise.
So,
KE1 = KE2 + PE
Substituting the equations for kinetic energy and potential energy:
(1/2) * m * v1^2 = (1/2) * m * v2^2 + m * g * h
Canceling out the common factor of mass and simplifying the equation:
v1^2 = v2^2 + 2 * g * h
Now we can solve for v2, the speed at the top of the rise:
v2^2 = v1^2 - 2 * g * h
v2^2 = (80 km/h)^2 - 2 * (9.8 m/s^2) * (10 m)
Converting km/h to m/s:
v2^2 = (80 m/s)^2 - 2 * (9.8 m/s^2) * (10 m)
Calculating:
v2^2 = 6400 m^2/s^2 - 196 m^2/s^2
v2^2 = 6204 m^2/s^2
Taking the square root:
v2 = √6204 m/s
v2 ≈ 78.7 m/s
Therefore, the car would have a speed of approximately 78.7 m/s at the top of the rise.