A 2.1 multiplied by 103-kg car starts from rest at the top of a 5.8-m-long driveway that is inclined at 22° with the horizontal. If an average friction force of 4.0 multiplied by 103 N impedes the motion, find the speed of the car at the bottom of the driveway.
Doing brazosport homework huh?
Whoa, a bit over 3 years late to this party but mind blown. I am actually doing Brazosport homework lmao. But these Physics problems are used by tons of schools, what are the chances?
To find the speed of the car at the bottom of the driveway, we can use the principles of physics, specifically the concepts of work, energy, and forces.
The first step is to find the net force acting on the car. This can be calculated using the force components along the incline.
The weight of the car can be calculated using the formula:
weight = mass * gravity
Since the mass of the car is given as 103 kg and the acceleration due to gravity is approximately 9.8 m/s², the weight of the car can be calculated as:
weight = 103 kg * 9.8 m/s² = 1009.4 N
Next, we need to find the component of the weight acting along the incline. This can be calculated using the equation:
force_parallel = weight * sin(angle)
where the angle is given as 22°.
force_parallel = 1009.4 N * sin(22°) = 359.46 N
The frictional force acting in the opposite direction can be represented as a negative quantity, since it opposes the motion. Therefore, the frictional force is -4.0 * 10^3 N.
The net force can be calculated by subtracting the frictional force from the force parallel to the incline:
net_force = force_parallel - frictional_force
= 359.46 N - (-4.0 * 10^3 N)
= 359.46 N + 4000 N
= 4359.46 N
Since the net force is along the incline, we can use this force to calculate the work done on the car as it moves down the incline using the formula:
work = net_force * distance
where the distance is given as 5.8 m.
work = 4359.46 N * 5.8 m = 25251.35 N·m or joules
According to the work-energy principle, the work done on the car equals the change in its kinetic energy:
work = ΔKE
Initially, the car is at rest, so its initial kinetic energy is zero. Therefore, the final kinetic energy is equal to the work done on the car.
KE_final = work = 25251.35 N·m or joules
Using the equation:
KE = 0.5 * mass * velocity^2
we can rearrange the equation to solve for velocity:
velocity = sqrt(2 * KE / mass)
velocity = sqrt(2 * 25251.35 N·m / 103 kg)
= sqrt(490.86 m²/s²)
≈ 22.14 m/s
Therefore, the speed of the car at the bottom of the driveway is approximately 22.14 m/s.