A 2110 kg car traveling at 10.7 m/s collides with a 2780 kg car that is initially at rest at the stoplight. The cars stick together and move 2.00 m before friction causes them to stop. Determine the coefficient of kinetic friction betwen the cars and the road, assuming that the negative acceleration is constant and that all wheels on both cars lock at the time of impact.
conservation of momentum
2110*10.7+0=(2110+2780)V
solve for Velocity V after colliding.
Energy in the moving cars=workby friction
1/2 (2100+2780)V^2=mu*(totalmass)g*distance
1/2 v^2/(g*distance)=mu
Thank you so much
To determine the coefficient of kinetic friction between the cars and the road, we need to follow these steps:
Step 1: Calculate the initial momentum of the first car.
The initial momentum (p1) of the first car is given by the formula:
p1 = m1 * v1
where m1 is the mass of the first car (2110 kg) and v1 is the initial velocity of the first car (10.7 m/s).
p1 = 2110 kg * 10.7 m/s
p1 = 22,537 kg·m/s
Step 2: Calculate the initial momentum of the second car.
Since the second car is initially at rest, its initial momentum (p2) is zero.
p2 = 0
Step 3: Calculate the total momentum after the collision.
Since the cars stick together and move as one after the collision, the total momentum (ptotal) is the sum of the initial momenta of both cars.
ptotal = p1 + p2
ptotal = 22,537 kg·m/s + 0
ptotal = 22,537 kg·m/s
Step 4: Determine the stopping distance and calculate the work done by friction.
Given the stopping distance (d = 2.00 m), we can calculate the work done by friction using the formula:
Work = Force * Distance * cosθ
In this case, the force is the force of friction, the distance is the stopping distance, and cosθ is 1 since the force of friction is in the opposite direction of motion.
Work = Force * Distance * cosθ = -μ * m * g * d
where μ is the coefficient of kinetic friction, m is the total mass of the two cars (2110 kg + 2780 kg = 4890 kg), g is the acceleration due to gravity (9.8 m/s^2), and d is the stopping distance.
The negative sign for the work is due to the friction force opposing the direction of motion.
Step 5: Calculate the work done by friction.
We can use the work-energy principle, which states that the work done by an external force (in this case, friction) is equal to the change in kinetic energy.
Therefore, the work done by friction is equal to the initial kinetic energy of the cars.
Initial kinetic energy = (1/2) * m * v^2
where m is the total mass of the two cars and v is the initial velocity.
Initial kinetic energy = (1/2) * (4890 kg) * (10.7 m/s)^2
Initial kinetic energy = 26,060.515 J
Step 6: Calculate the coefficient of kinetic friction.
Finally, we can set the work done by friction equal to the initial kinetic energy and solve for the coefficient of kinetic friction (μ).
-μ * m * g * d = Initial kinetic energy
-μ * (4890 kg) * (9.8 m/s^2) * (2.00 m) = 26,060.515 J
Solving for μ:
μ = -(26,060.515 J) / [(4890 kg) * (9.8 m/s^2) * (2.00 m)]
Calculating this expression will give us the coefficient of kinetic friction between the cars and the road.