A compact car, mass 875 kg, moving south at 16 m/s, is struck by a full-sized car, mass 1660 kg, moving east at 12 m/s. The two cars stick together, and momentum is conserved in the collision. The wreck skids along the ground and comes to a stop. The coefficient of kinetic friction while the wreck is skidding is 0.60. Assume that the acceleration is constant. How far does the wreck skid?
To find the distance the wreck skids, we need to determine the deceleration of the system and use it to calculate the distance traveled.
First, let's find the initial momentum of the system before the collision. Momentum is calculated as the product of mass and velocity, so:
Momentum_initial = (mass_compact_car * velocity_compact_car) + (mass_full_sized_car * velocity_full_sized_car)
Mass_compact_car = 875 kg
Velocity_compact_car = 16 m/s
Mass_full_sized_car = 1660 kg
Velocity_full_sized_car = 12 m/s
Momentum_initial = (875 kg * 16 m/s) + (1660 kg * 12 m/s)
Momentum_initial = 14000 kg*m/s + 19920 kg*m/s
Momentum_initial = 33920 kg*m/s
Since momentum is conserved in the collision, the final momentum of the system is zero (because the wreck comes to a stop):
Momentum_final = 0
The change in momentum (Δp) is then equal to the initial momentum:
Δp = Momentum_initial - Momentum_final
Δp = 33920 kg*m/s - 0
Δp = 33920 kg*m/s
Next, we need to find the force of friction opposing the motion of the wreck. The force of friction can be calculated as the product of the coefficient of kinetic friction (μ) and the normal force (N). Since the wreck is skidding, the normal force is equal to the weight of the system:
Normal force = (mass_compact_car + mass_full_sized_car) * gravity
Gravity = 9.8 m/s^2
Normal force = (875 kg + 1660 kg) * 9.8 m/s^2
Normal force = 2535 kg * 9.8 m/s^2
Normal force = 24813 N
Force of friction = μ * Normal force
Force of friction = 0.60 * 24813 N
Force of friction = 14887.8 N
Now we can use Newton's second law, which states that force (F) is equal to mass (m) times acceleration (a):
Force of friction = mass * deceleration
Deceleration = Force of friction / mass
Deceleration = 14887.8 N / (875 kg + 1660 kg)
Deceleration = 14887.8 N / 2535 kg
Deceleration = 5.87 m/s^2
Finally, we can use the kinematic equation to find the distance (d) the wreck skids:
v_f^2 = v_i^2 + 2ad
Since the initial velocity (v_i) is 0 (the wreck comes to a stop) and the final velocity (v_f) is also 0, the equation simplifies to:
0 = 0 + 2ad
Solving for d:
d = 0 / (2 * deceleration)
d = 0 / (2 * 5.87 m/s^2)
d = 0
Therefore, the wreck skids a distance of 0 meters.