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A 2.06×10³ kilogram car starts from rest and coasts down from the top of a 4.35 meter long driveway that is sloped at an angle of 21.3o with the horizontal. if an average friction force of 3.93×10³ newtons impedes the motion of the car, calculate the speed of the car at the bottom of the driveway.

User Allen
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To calculate the speed of the car at the bottom of the driveway, we can use the principle of conservation of energy.

The initial potential energy (PE) of the car at the top of the driveway is given by:

PE = mgh

where m is the mass of the car (2.06×10³ kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the driveway (4.35 m).

PE = (2.06×10³ kg)(9.8 m/s²)(4.35 m)
= 8.96×10⁴ J

The work done by the friction force is given by:

Work = Force x Distance

The friction force is given as 3.93×10³ N, and the distance is 4.35 m.

Work = (3.93×10³ N)(4.35 m)
= 1.71×10⁴ J

The final kinetic energy (KE) of the car at the bottom of the driveway is equal to the initial potential energy minus the work done by friction:

KE = PE - Work
= 8.96×10⁴ J - 1.71×10⁴ J
= 7.25×10⁴ J

The kinetic energy is given by:

KE = 1/2 mv²

where v is the velocity of the car at the bottom of the driveway.

Rearranging the equation, we have:

v² = 2KE / m
v² = (2)(7.25×10⁴ J) / (2.06×10³ kg)
v² = 7.05×10⁴ J / (2.06×10³ kg)
v² = 34.22 m²/s²

Taking the square root of both sides, we get:

v = √(34.22 m²/s²)
v ≈ 5.85 m/s

Therefore, the speed of the car at the bottom of the driveway is approximately 5.85 m/s.
User Kimarley
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