To solve these problems, we can use the principles of kinematics and Newton's laws of motion. Let's break it down step-by-step:
a) How fast will the car be moving at point B?
Step 1: Determine the forces acting on the car.
- Gravitational force: The weight of the car is given by W = m * g, where m is the mass of the car (1500 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- Friction force: The friction force opposing motion on the inclined plane is given as 400 N.
Step 2: Resolve the gravitational force into components.
The weight of the car can be split into two components: one parallel to the inclined plane (W_parallel = W * sinθ) and one perpendicular to the inclined plane (W_perpendicular = W * cosθ), where θ is the angle of inclination (10 degrees).
Step 3: Calculate the net force acting on the car.
The net force is the difference between the parallel component of weight and the friction force: Net force = W_parallel - Friction = (W * sinθ) - Friction.
Step 4: Calculate the acceleration of the car.
Using Newton's second law (F = m * a), we can find the acceleration: Net force = m * a. Rearranging the equation gives us:
a = (Net force) / m.
Step 5: Calculate the final velocity at point B.
Since the car starts from rest, we can use the equation of motion (v^2 = u^2 + 2 * a * s) to find the final velocity at point B. Here, u is the initial velocity (0 m/s), and s is the displacement.
Now, let's put it all together and solve for the final velocity at point B:
Step 1:
Weight of the car, W = m * g = 1500 kg * 9.8 m/s^2 = 14700 N.
Step 2:
W_parallel = W * sinθ = 14700 N * sin(10°) = 2547.9 N.
W_perpendicular = W * cosθ = 14700 N * cos(10°) = 14511.0 N.
Step 3:
Net force = W_parallel - Friction = 2547.9 N - 400 N = 2147.9 N.
Step 4:
Acceleration, a = (Net force) / m = 2147.9 N / 1500 kg = 1.43 m/s^2.
Step 5:
Using the equation v^2 = u^2 + 2 * a * s and assuming u = 0:
v^2 = 0^2 + 2 * 1.43 m/s^2 * s.
v^2 = 2.86 m/s^2 * s.
Since we want the speed, we can rearrange the equation to solve for v:
v = √(2.86 m/s^2 * s).
Therefore, the car will be moving at a speed of √(2.86 m/s^2 * s) at point B.
b) How far from point C will it travel before coming to rest?
To find this, we need to determine the total distance the car will travel after coming to rest. This can be found by calculating the distance traveled on the inclined plane (d1) and then adding it to the distance traveled on the horizontal surface (d2).
Step 1: Calculate the distance traveled on the inclined plane (d1).
Using the equation of motion v^2 = u^2 + 2 * a * s, with v = 0 m/s at the end point:
0^2 = (v^2) - 2 * a * d1.
d1 = (v^2) / (2 * a).
Step 2: Calculate the distance traveled on the horizontal surface (d2).
On the horizontal surface, the only force acting on the car is the friction force of 600 N.
Using the equation of motion v^2 = u^2 + 2 * a * s, with v = 0 m/s at the end point:
0^2 = (v^2) - 2 * (Friction / m) * d2.
d2 = (v^2) / (2 * (Friction / m)).
Step 3: Calculate the total distance traveled.
Total distance = d1 + d2.
Now, let's calculate the values:
Step 1:
d1 = (v^2) / (2 * a) = (0 m/s)^2 / (2 * 1.43 m/s^2) = 0 m.
Step 2:
d2 = (v^2) / (2 * (Friction / m)) = (0 m/s)^2 / (2 * (600 N / 1500 kg)) = 0 m.
Step 3:
Total distance = d1 + d2 = 0 m + 0 m = 0 m.
Therefore, the car will travel a total distance of 0 meters before coming to rest from point C.