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A car is on an icy driveway inclined at an angle theta = 20.0°. The coefficient of kinetic friction between the tire and ice is 0.15. If the length of the driveway is 25.0m and the car starts from rest at the top, what is its speed at the end of the driveway?

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To find the speed of the car at the end of the driveway, we can use principles of physics, specifically Newton's laws and the concept of work and energy.

First, let's break down the forces acting on the car. We have the force of gravity pulling the car downwards and the force of kinetic friction opposing the motion. The force of gravity can be resolved into two components: one parallel to the inclined surface and one perpendicular to it.

The component of gravity parallel to the incline can be calculated using the equation:

F_parallel = m * g * sin(theta)

Here, m represents the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s^2), and theta is the angle of inclination (20.0°).

The force of kinetic friction can be calculated using:

F_friction = coefficient of friction * F_normal

F_normal is the normal force exerted on the car, which is equal to the component of gravity perpendicular to the incline:

F_normal = m * g * cos(theta)

Now that we have the forces, we need to calculate the work done on the car. The work done by the net force on an object is equal to the change in its kinetic energy. In this case, the work done by the net force is given by:

Work = force parallel * distance

Since the car starts from rest, the initial kinetic energy is zero. Therefore, the work done on the car is equal to its final kinetic energy:

Work = (1/2) * m * v^2

Setting these equations equal to each other, we can solve for the final velocity (v):

(1/2) * m * v^2 = (m * g * sin(theta) - coefficient of friction * m * g * cos(theta)) * distance

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

theta = 20.0°
coefficient of kinetic friction = 0.15
distance = 25.0 m

Plugging in these values and rearranging the equation, we can solve for v:

v = sqrt((2 * (m * g * sin(theta) - coefficient of friction * m * g * cos(theta)) * distance) / m)

Remember to convert the angle theta from degrees to radians before performing calculations.

I hope this explanation helps you understand how to calculate the speed of the car at the end of the driveway. If you have any further questions or need assistance with any other topic, feel free to ask!
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