A 4.2 × 10^3

kg car accelerates from rest at
the top of a driveway that is sloped at an angle of 18.4◦
with the horizontal. An average
frictional force of 4.1×10^3 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/s^2.
What is the length of the driveway?
Answer in units of m

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To find the length of the driveway, we can use the equations of motion and solve for the unknown variable.

Let's break down the problem and gather the given information:

Mass of the car (m) = 4.2 × 10^3 kg
Angle of the slope (θ) = 18.4 degrees
Frictional force (F_friction) = 4.1 × 10^3 N
Speed at the bottom of the driveway (v) = 4.9 m/s
Acceleration due to gravity (g) = 9.81 m/s^2

First, let's resolve the weight of the car (mg) parallel and perpendicular to the slope:

Weight parallel to the slope (mg_parallel) = m * g * sin(θ)

Weight perpendicular to the slope (mg_perpendicular) = m * g * cos(θ)

Next, let's calculate the net force acting on the car parallel to the slope:

Net force parallel to the slope (F_net_parallel) = mg_parallel - F_friction

Since we want to find the length of the driveway, we need to calculate the acceleration of the car. We can use the equation of motion:

F_net_parallel = m * a

From this equation, we can solve for acceleration (a):

a = F_net_parallel / m

Now, let's substitute the values into the equation:

a = (mg_parallel - F_friction) / m

Finally, we can use this acceleration value to find the length of the driveway using the equation of motion:

v^2 = u^2 + 2as

Since the car starts from rest at the top of the driveway, the initial velocity (u) is 0.

Simplifying the equation, we have:

s = v^2 / (2a)

Now, we can substitute the values into the equation and solve for the length of the driveway (s):

s = 4.9^2 / (2 * a)

To find the value of 'a', we can substitute it with the previously calculated acceleration:

s = 4.9^2 / (2 * [(mg_parallel - F_friction) / m])

Now, let's plug in the values and calculate:

s = 4.9^2 / (2 * [(4.2 × 10^3 kg * 9.81 m/s^2 * sin(18.4◦) - 4.1 × 10^3 N) / (4.2 × 10^3 kg)])

By evaluating this expression, we can find the length of the driveway.