To find the speed of the second car when it hit the stopped car, we can use the principles of conservation of momentum and the equations of motion.
Let's assume the speed of the first car before it stopped was v1, and the speed of the second car before it hit the stopped car as v2.
Step 1: Find the initial momentum of both cars.
The momentum of an object is given by the equation: momentum = mass × velocity.
The momentum of the first car (which stopped) can be calculated as:
p1 = m1 × v1
where m1 = mass of the first car = 1875 kg, and v1 = velocity of the first car before it stopped.
The momentum of the second car can be calculated as:
p2 = m2 × v2
where m2 = mass of the second car = 2135 kg, and v2 = velocity of the second car before it hit the stopped car.
Step 2: Calculate the final momentum of the system.
Since the two cars lock together and move as one after the collision, their total momentum is conserved.
The final momentum is given by:
p_final = m1 × v_final
where v_final = final velocity of both cars after the collision.
Step 3: Calculate the change in momentum.
The change in momentum is given by the equation:
Δp = p_final - p2
where Δp = change in momentum.
Step 4: Calculate the frictional force.
The frictional force can be calculated using the equation:
frictional force = coefficient of friction × normal force
In this case, the normal force is equal to the weight of the two cars, which can be calculated as:
normal force = (m1 + m2) × g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 5: Convert the frictional force into acceleration.
The frictional force can be converted into acceleration using Newton's second law of motion:
frictional force = mass × acceleration
where mass = total mass of the two cars = m1 + m2.
Step 6: Calculate the deceleration of the second car.
Since the second car is braking and coming to a stop, its acceleration is in the opposite direction of its initial velocity. Therefore, the deceleration can be calculated as the negative of the acceleration calculated in step 5.
Step 7: Calculate the time taken for the second car to come to a stop.
We can use the equation of motion: v_final^2 = v_initial^2 + 2 × acceleration × distance,
where v_final = 0 (as the car comes to a stop),
v_initial = v2 (initial velocity of the second car),
acceleration = deceleration calculated in step 6, and
distance = 4.58 m.
Solving this equation will give us the time taken for the second car to come to a stop.
Step 8: Calculate the final velocity of both cars.
Using the equation of motion: v_final = v_initial + acceleration × time,
where v_final = final velocity of both cars after the collision,
v_initial = v2 (initial velocity of the second car),
acceleration = deceleration calculated in step 6, and
time = time taken for the second car to come to a stop (calculated in step 7).
Solving this equation will give us the final velocity of both cars.
This final velocity will be the speed of the second car when it hit the stopped car.