A 2.9 × 103 kg car accelerates from rest at

the top of a driveway that is sloped at an
angle of 16.0� with the horizontal. An average
frictional force of 4.2×103 N impedes the car’s
motion so that the car’s speed at the bottom
of the driveway is 4.6 m/s.
The acceleration of gravity is 9.81 m/s2 .
What is the length of the driveway?

x = unknown length of driveway

distance it fell: h = x sin 16

potential energy lost = m g h = m g x sin 16

Ke gained = (m/2)(4.6)^2

difference = work done by friction = 4.3*10^3 * x

so
m g x sin 16 = (m/2)4.6^2 + 4.3*10^3 x

To find the length of the driveway, we can use the equations of motion for linear motion with constant acceleration. The given information includes the mass of the car (2.9 × 10^3 kg), the angle of the driveway (16.0°), the frictional force (4.2 × 10^3 N), and the speed of the car at the bottom of the driveway (4.6 m/s). We also know the acceleration due to gravity (9.81 m/s^2).

First, let's resolve the gravitational force acting on the car parallel and perpendicular to the slope. The gravitational force acting downwards can be calculated by multiplying the mass of the car (m) by the acceleration due to gravity (g):

Gravitational force (Fg) = m * g

Next, we need to find the component of the weight that is parallel to the slope, which is given by:

Weight component parallel to the slope (Fp) = Fg * sin(θ)

where θ is the angle of the driveway (16.0°).

Since the car is on a slope, it experiences a force due to gravity acting parallel to the slope. We can find the magnitude of this parallel force (Fp) by substituting the values of Fg and θ into the equation.

Next, we need to calculate the net force acting on the car. The net force (Fnet) is the difference between the parallel component of the weight (Fp) and the frictional force (Ff):

Net force (Fnet) = Fp - Ff

where Ff is the frictional force (4.2 × 10^3 N).

Using Newton's second law of motion (F = ma), we can find the acceleration (a) of the car:

Acceleration (a) = Fnet / m

Finally, we can use the motion equation v^2 = u^2 + 2as, where u is the initial velocity (0 m/s) and v is the final velocity (4.6 m/s), to find the length of the driveway (s):

Length of the driveway (s) = (v^2 - u^2) / (2a)

Using the given values and the calculated values, we can substitute them into the equation to find the length of the driveway.