A car with engine trouble slides 85 meters up along a 14 degrees ramp before coming to a stop. If the coefficient of friction between the tires and the road is 0.9, what was the velocity of the car at the bottom of the ramp?
I get 32.43 meters / second
is this correct?
initial KE=workdonebyfriction+final PE
1/2 m v^2= mg*mu*cos14*85 + mg*85sin14
solve for v.
can you explain how you derive that equation?
we haven't learned about potential energy yet? is there another way to solve this?
To solve this problem, we can use the concept of mechanical energy conservation. At the bottom of the ramp, the car's mechanical energy is entirely in the form of kinetic energy. At the top of the ramp, the car's mechanical energy is in the form of potential energy (due to its height) and kinetic energy (due to its velocity).
First, let's calculate the potential energy at the top of the ramp. The potential energy formula is:
Potential energy = mass x gravitational acceleration x height
Given that the height is the vertical displacement of the car, which is given as 85 meters up, and the gravitational acceleration is approximately 9.8 m/s^2:
Potential energy = mass x 9.8 x 85
Next, let's calculate the kinetic energy at the bottom of the ramp. The kinetic energy formula is:
Kinetic energy = (1/2) x mass x velocity^2
Since we want to find the velocity at the bottom of the ramp, we need to rearrange the formula:
Velocity^2 = (2 x kinetic energy) / mass
Now we know that at the bottom of the ramp, there is no potential energy, so the total mechanical energy is equal to the kinetic energy:
Potential energy at the top = Kinetic energy at the bottom
Mass x 9.8 x 85 = (1/2) x mass x velocity^2
Simplifying this equation, we can cancel out "mass" on both sides:
9.8 x 85 = (1/2) x velocity^2
Rearranging the equation to solve for velocity:
velocity^2 = (9.8 x 85) / 0.5
velocity^2 = 1666
Taking the square root of both sides to solve for velocity:
velocity ≈ 40.82 meters/second
Therefore, your calculation of 32.43 meters/second is incorrect. The correct velocity at the bottom of the ramp is approximately 40.82 meters/second.