A 3.5 × 103
kg car accelerates from rest at
the top of a driveway that is sloped at an angle of 16.9◦ with the horizontal. An average frictional force of 4.1×103 N impedes the car’s
motion so that the car’s speed at the bottom of the driveway is 4.2 m/s.
The acceleration of gravity is 9.81 m/s^2.
What is the length of the driveway?
Answer in units of m
What do you say?
To find the length of the driveway, we can use the equations of motion for linear motion. There are two main components of motion that we need to consider: the car's acceleration down the slope and the opposing force of friction.
First, let's calculate the car's acceleration down the slope. We can use the component of gravity down the slope, which can be determined by multiplying the acceleration due to gravity (9.81 m/s^2) by the sine of the slope angle (16.9°):
Acceleration down the slope = 9.81 m/s^2 * sin(16.9°)
Next, let's calculate the net force acting on the car. The net force is the difference between the force of gravity down the slope and the force of friction:
Net Force = (mass of the car) * (acceleration down the slope) - (frictional force)
Since the car is at rest at the top of the driveway, the initial velocity is 0 m/s. The final velocity at the bottom of the driveway is given as 4.2 m/s. We can use the following kinematic equation to relate the net force, mass, and final velocity:
Net Force = (mass of the car) * (final velocity)^2 / (2 * distance)
Here, distance refers to the length of the driveway that we want to find.
Now, we can solve for the length of the driveway:
(mass of the car) * (acceleration down the slope) - (frictional force) = (mass of the car) * (final velocity)^2 / (2 * distance)
Rearranging the equation and solving for the distance, we get:
distance = (mass of the car) * (final velocity)^2 / [(2 * ((mass of the car) * (acceleration down the slope) - (frictional force)))]
Plugging in the given values:
mass of the car = 3.5 × 10^3 kg
acceleration down the slope = 9.81 m/s^2 * sin(16.9°)
frictional force = 4.1 × 10^3 N
final velocity = 4.2 m/s
distance = (3.5 × 10^3 kg) * (4.2 m/s)^2 / [2 * ((3.5 × 10^3 kg) * (9.81 m/s^2 * sin(16.9°)) - (4.1 × 10^3 N))]
Evaluating this expression will give us the length of the driveway.