A 2.1 103-kg car starts from rest at the top of a 5.9-m-long driveway that is inclined at 19° with the horizontal. If an average friction force of 4.0 103 N impedes the motion, find the speed of the car at the bottom of the driveway.
m/s
Hooke's law describes a certain light spring of unstretched length 32.0 cm. When one end is attached to the top of a door frame and a 5.40-kg object is hung from the other end, the length of the spring is 46.5 cm.
(a) Find its spring constant.
kN/m
(b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 160 N. Find the length of the spring in this situation.
To find the speed of the car at the bottom of the driveway, we can use the concept of mechanical energy.
First, we need to calculate the gravitational potential energy (GPE) of the car at the top of the driveway. The formula for GPE is given by:
GPE = m * g * h
where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the driveway.
Given:
mass of car (m) = 2.1 × 10^3 kg
height of driveway (h) = 5.9 m
acceleration due to gravity (g) = 9.8 m/s^2
Calculating the GPE:
GPE = (2.1 × 10^3 kg) * (9.8 m/s^2) * (5.9 m)
= 116179 N·m (or Joules)
Next, we need to take into account the work done by the friction force. The work done by a force is given by the product of the force and the distance over which it acts. In this case, the distance is the length of the driveway (5.9 m) and the force is the friction force (4.0 × 10^3 N). Since the work done by friction is negative (it opposes motion), the equation becomes:
Work done by friction (W) = -F * d
= -(4.0 × 10^3 N) * (5.9 m)
= -23600 N·m (or Joules)
Now, we can find the kinetic energy (KE) of the car at the bottom of the driveway using the conservation of mechanical energy:
KE = GPE + W
= 116179 N·m + (-23600 N·m)
= 92579 N·m (or Joules)
Finally, we can use the kinetic energy formula to find the speed (v) of the car at the bottom of the driveway:
KE = 1/2 * m * v^2
92579 N·m = 0.5 * (2.1 × 10^3 kg) * v^2
Simplifying the equation:
v^2 = (2 * 92579 N·m) / (2.1 × 10^3 kg)
v^2 = 87599.52 m^2/s^2
v ≈ √(87599.52) ≈ 296.08 m/s
Therefore, the speed of the car at the bottom of the driveway is approximately 296.08 m/s.