A 3.1 × 103 kg car accelerates from rest at the top of a driveway that is sloped at an angle of 19.5◦ 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 4.9 m/s.

The acceleration of gravity is 9.81 m/s2 . What is the length of the driveway

Wc = m*g = 3100kg * 9.8N/kg = 30,380 N.=

Wt. of car.

Fp = 30,380*sin19.5 = 10,141 N. = Force parallel to incline.
Fv = 30,380*cos19.5 = 28,637 N. = Force
perpendicular to incline.

Fk=4400 N. = Force of kinetic friction.

Fp-Fk = m*a
a=(Fp-Fk)/m
a = (10,141-4400)/3100=1.852 m/s^2.

L = (V^2-Vo^2)/2a=(4.9^2-0)/3.704=6.48 m

To find the length of the driveway, we can use the kinematic equation that relates distance, initial velocity, final velocity, and acceleration:

v^2 = u^2 + 2as

Where:
v = final velocity
u = initial velocity (which is 0 m/s in this case since the car starts from rest)
a = acceleration
s = distance

We are given:
u = 0 m/s
v = 4.9 m/s
a = ?

First, let's calculate the acceleration of the car. To do this, we need to consider the external forces acting on the car. We have the gravitational force pulling the car down the slope and the frictional force opposing the motion.

The component of the weight pulling the car down the slope is given by:

F_gravity = m * g * sinθ

Where:
m = mass of the car = 3.1 × 10^3 kg
g = acceleration due to gravity = 9.81 m/s^2
θ = angle of the slope = 19.5 degrees

F_gravity = (3.1 × 10^3 kg) * (9.81 m/s^2) * sin(19.5 degrees)

Next, we can calculate the net force acting on the car by subtracting the frictional force from the gravitational force:

F_net = F_gravity - F_friction

Where:
F_gravity = gravitational force calculated above
F_friction = 4.4 × 10^3 N (frictional force given in the question)

F_net = F_gravity - F_friction

Now, we can find the acceleration using Newton's second law:

F_net = m * a

Substitute the value of the net force (F_net) obtained above and the mass (m) given in the question into the equation, and solve for acceleration (a).

Now that we have the acceleration, we can use the kinematic equation to find the distance (s).

v^2 = u^2 + 2as

Substitute the values of v (4.9 m/s), u (0 m/s), and a (acceleration obtained above) into the equation, and solve for s. The resulting value of s will be the length of the driveway.