A car with mass of 1875kg is traveling along a country road when the driver sees a deer dart out onto the road. The driver slams on the brakes manages to stop before hitting the deer. The driver of a second car with 2135kg is driving too close and does not see the deer. When the driver realizes that the car ahead is stopping, he hits the brakes but is unable to stop. The cars lock together and skid another 4.58m. All of the motion is along a straight line. If the coefficient of friction between the dry concrete and rubber tires is 0.750, what was the speed of the second car when it hit the stopped car?
First solve for the speed of the cars locked together, V', using the fact that the kinetic energy lost equals work done against friction during the skid.
(1/2)(M+m)V'^2 = (M+m)(Uk)*g*X
V'^2 = 2*0.750*9.8*4.58 = 63.3 m/s^2
Then apply conservation of momentum to determine the speed of the faster car that caused the collision, V.
M V = (M+m)*V'
Solve for V
45m/s
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.
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 work-energy principle.
First, let's find the initial speed of the first car before it stopped. Since it was able to stop without hitting the deer, the net force acting on it must be equal to the force of friction opposing its motion:
Friction force = μ * Normal force
Where:
μ = coefficient of friction
Normal force = mass * gravity
The friction force will act in the opposite direction of motion, so we can find the net force by subtracting the friction force from the forward force:
Net force = forward force - friction force
Now, the work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the net force on the first car is equal to the initial kinetic energy of the car:
Work = change in kinetic energy
Work = force * distance (since force and displacement are in the same direction)
The work done by the net force is equal to the force of friction multiplied by the distance the car traveled before stopping:
Work = (μ * Normal force) * distance
Since the car came to a stop, the initial kinetic energy becomes zero. Therefore:
(μ * Normal force) * distance = 0
Now let's consider the collision between the two cars. The total momentum before the collision is equal to the total momentum after the collision. The momentum of an object is given by the product of its mass and velocity:
Initial momentum = final momentum
For the first car:
Initial momentum = mass of the first car * velocity of the first car (since it is at rest)
Final momentum = 0 (since it stopped)
For the second car:
Initial momentum = mass of the second car * initial velocity of the second car
Final momentum = (mass of the first car + mass of the second car) * final velocity (since the cars lock together and move together)
We can rearrange the equation to find the final velocity:
final velocity = (mass of the second car * initial velocity of the second car) / (mass of the first car + mass of the second car)
Substitute the given values into the equation, where:
mass of the first car = 1875 kg
mass of the second car = 2135 kg
initial velocity of the second car = unknown
final velocity = unknown
Now we have an equation to find the final velocity:
final velocity = (2135 kg * initial velocity of the second car) / (1875 kg + 2135 kg)
Finally, we have the equation to solve for the final velocity.