Question

A 6.0 × 103 kg car accelerates from rest at the top of a driveway that is sloped at an angle of 19.3 ◦ with the horizontal. An average frictional force of 4.4×103 N impedes the car’s motion so that the car’s speed at the bottom of the driveway is 4.3 m/s. The acceleration of gravity is 9.81 m/s 2 . What is the length of the driveway? Answer in units of m.

Answer 1

To find the length of the driveway, we can use the equations of motion. Calculate the acceleration using trigonometry, find the net force, and use the equation v^2 = u^2 + 2as to determine the distance.

Step-by-step explanation:

To find the length of the driveway, we can use the equations of motion. The acceleration of the car can be found by considering the forces acting on it. The force of gravity can be resolved into two components: one along the slope and one perpendicular to the slope. The force along the slope, which is responsible for the acceleration of the car, can be found using trigonometry. The net force acting on the car is the force along the slope minus the frictional force. We can then use the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance traveled, to find the length of the driveway.

Let's break the problem down step by step:

1. Calculate the acceleration of the car along the slope using trigonometry: acceleration = g * sin(angle).
2. Find the net force acting on the car: net force = m * acceleration - frictional force.
3. Use the equation v^2 = u^2 + 2as to find the length of the driveway: distance = (v^2 - u^2) / (2 * acceleration).

Substituting the given values into the equations will give you the length of the driveway.

Answer 2

s = 3.68 m

Step-by-step explanation:

given,

mass of the car = 6 x 10³ Kg

angle of slope with horizontal = 19.3°

Average frictional force = 4.4×10³ N

Speed of the car at the bottom = 4.3 m/s

acceleration due to gravity = 9.81 m/s²

length of driveway = ?

first calculating the net force acting on the car

F = mg sin θ - f

(mg sin θ) is the weight component of the body

(f) is the frictional force which is acting opposite to the car

F = 6000 x 9.81 x sin 19.3° - 4400

F = 19454 - 4400

F = 15054 N

we know,

Force = mass x acceleration

15054 = 6000 x a

a = 2.51 m/s²

using formula

v² = u² + 2 as

4.3² = 0² + 2 x 2.51 x s

s = 3.68 m

the length of the driveway = s = 3.68 m