A 2.10 103 kg car starts from rest at the top of a 5.8 m long driveway that is sloped at 25° with the horizontal. If an average friction force of 4.0 103 N impedes the motion, find the speed of the car at the bottom of the driveway.
To find the speed of the car at the bottom of the driveway, we can use the principles of physics along with some equations.
Step 1: Break down the given information:
- Mass of the car (m): 2.10 × 10^3 kg
- Length of the driveway (d): 5.8 m
- Slope angle (θ): 25°
- Friction force (f): 4.0 × 10^3 N
Step 2: Calculate the force component parallel to the slope:
The force component parallel to the slope can be calculated using the formula: Fparallel = m * g * sin(θ), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fparallel = (2.10 × 10^3 kg) * (9.8 m/s^2) * sin(25°)
Fparallel = (2.10 × 10^3) * (9.8) * (0.423)
Fparallel ≈ 8.156 × 10^3 N
Step 3: Calculate the net force:
The net force acting on the car is the force parallel to the slope minus the friction force.
Net force (Fnet) = Fparallel - f
Fnet = 8.156 × 10^3 N - 4.0 × 10^3 N
Fnet = 4.156 × 10^3 N
Step 4: Calculate the acceleration:
The acceleration (a) of the car can be calculated using Newton's second law: Fnet = m * a.
a = Fnet / m
a = (4.156 × 10^3 N) / (2.10 × 10^3 kg)
a ≈ 1.98 m/s^2
Step 5: Calculate the final velocity (speed):
Using the equation for motion along an inclined plane, we can calculate the final velocity (v) of the car at the bottom of the driveway.
v^2 = u^2 + 2as
Where u is the initial velocity (which is zero because the car starts from rest), a is the acceleration calculated in Step 4, and s is the distance traveled (which is the length of the driveway).
v^2 = (0)^2 + 2 * (1.98 m/s^2) * (5.8 m)
v^2 ≈ 22.92 m^2/s^2
v ≈ √(22.92 m^2/s^2)
v ≈ 4.79 m/s
Therefore, the speed of the car at the bottom of the driveway is approximately 4.79 m/s.