A truck of mass 9M is moving at a speed v = 15 km/h when it collides head on with a parked car of mass M. Spring mounted bumpers ensure that the collision is essentially elastic. Find the final velocities of each vehicle. Repeat for the case of the car moving at speed v= 15 km/h and colliding with the stationary truck. What fraction of the incident vehicle’s kinetic energy is transferred in each case?
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's start with the first case where the truck is moving at speed v and collides with the parked car.
1. Conservation of momentum:
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. The momentum (p) of an object is given by its mass (m) multiplied by its velocity (v).
Before the collision:
Truck momentum = 9M * v
Car momentum = 0 (as it is parked)
After the collision:
Let the final velocities of the truck and car be Vt and Vc, respectively.
Truck momentum = 9M * Vt
Car momentum = M * Vc
So, we have the equation: 9M * v = 9M * Vt + M * Vc ---(Equation 1)
2. Conservation of kinetic energy:
According to the conservation of kinetic energy, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
The kinetic energy (KE) of an object is given by (1/2) * mass * (velocity)^2.
Before the collision:
Truck kinetic energy = (1/2) * 9M * v^2
Car kinetic energy = 0 (as it is parked)
After the collision:
Truck kinetic energy = (1/2) * 9M * Vt^2
Car kinetic energy = (1/2) * M * Vc^2
So, we have the equation: (1/2) * 9M * v^2 = (1/2) * 9M * Vt^2 + (1/2) * M * Vc^2 ---(Equation 2)
Now, let's solve these equations simultaneously to find the final velocities of the truck and car.
Solving Equation 1 for Vt:
Vt = (9*v - Vc)/9
Substituting this value of Vt into Equation 2:
(1/2) * 9M * v^2 = (1/2) * 9M * ((9*v - Vc)/9)^2 + (1/2) * M * Vc^2
Simplifying the equation:
9M * v^2 = 9M * ((9*v - Vc)/9)^2 + M * Vc^2
Solving this equation will give us the final velocity Vc of the car. Once we have Vc, we can substitute it back into the equation for Vt to find the final velocity of the truck, Vt.
Now, let's move on to the second case where the car is moving at speed v and collides with the stationary truck.
Following the same process as above, we can set up the conservation equations and solve them simultaneously to find the final velocities Vt and Vc.
1. Conservation of momentum:
Before the collision:
Truck momentum = 0 (as it is stationary)
Car momentum = M * v
After the collision:
Truck momentum = 9M * Vt
Car momentum = M * Vc
So, we have the equation: M * v = 9M * Vt + M * Vc ---(Equation 3)
2. Conservation of kinetic energy:
Before the collision:
Truck kinetic energy = 0 (as it is stationary)
Car kinetic energy = (1/2) * M * v^2
After the collision:
Truck kinetic energy = (1/2) * 9M * Vt^2
Car kinetic energy = (1/2) * M * Vc^2
So, we have the equation: (1/2) * M * v^2 = (1/2) * 9M * Vt^2 + (1/2) * M * Vc^2 ---(Equation 4)
Solving Equations 3 and 4 simultaneously will give us the final velocities Vt and Vc of the truck and car, respectively.
To find the fraction of the incident vehicle's kinetic energy transferred in each case, we can calculate the ratio of the kinetic energy after the collision to the initial kinetic energy. For example, in the first case:
Fraction of kinetic energy transferred = (Kinetic energy of the truck after collision + Kinetic energy of the car after collision) / Initial kinetic energy of the truck
Similarly, we can calculate the fraction of kinetic energy transferred in the second case.
Note: To get the exact numerical values, you will need to substitute the given values for M, v, and solve the equations accordingly.