A 3.5 × 103
kg car accelerates from rest at
the top of a driveway that is sloped at an angle of 16.9◦ with the horizontal. An average frictional force of 4.1×103 N impedes the car’s
motion so that the car’s speed at the bottom of the driveway is 4.2 m/s.
The acceleration of gravity is 9.81 m/s^2.
What is the length of the driveway?
Answer in units of m
the net acceleration is
___ g sin(16.9º) - (4.1×103 N / 3.5 × 103
kg)
divide the speed at the bottom by the acceleration to get the time for the trip
the average speed is (0 + 4.2) / 2
speed times time is distance (the length of the driveway)
To find the length of the driveway, we can use the principles of Newton's laws of motion and kinematics equations.
First, let's analyze the forces acting on the car:
1) The force of gravity acting on the car is given by the product of its mass and the acceleration due to gravity: Fg = m * g, where m = 3.5 × 10^3 kg and g = 9.81 m/s^2.
2) The component of the force of gravity that acts along the slope of the driveway is given by Fg_parallel = Fg * sin(theta), where theta = 16.9°.
3) The frictional force acts opposite to the direction of motion and has a constant magnitude of 4.1 × 10^3 N.
The net force acting on the car is the vector sum of the force of gravity along the slope and the frictional force:
Net force = Fg_parallel - Frictional force
According to Newton's second law of motion, F_net = mass * acceleration. In this case, the acceleration is the acceleration of the car as it moves down the driveway.
Now we can set up the equation:
Fg_parallel - Frictional force = mass * acceleration
Solving for acceleration:
acceleration = (Fg_parallel - Frictional force) / mass
Next, we can use the kinematic equation to relate the acceleration, initial velocity, final velocity, and distance:
v^2 = u^2 + 2ax
where:
v = final velocity = 4.2 m/s,
u = initial velocity = 0 m/s (since the car starts from rest),
a = acceleration,
x = distance (length of the driveway).
Rearranging the equation:
x = (v^2 - u^2) / (2a)
Now we can substitute the values into the equation and calculate the length of the driveway:
x = (4.2^2 - 0^2) / (2 * acceleration)
Note: We already calculated the acceleration using the first equation.
Once you calculate the value of x, you will have the length of the driveway in meters.