A 1,225-kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a 8,500-kg truck moving in the same direction at 20.0 m/s. The velocity of the car right after the collision is 18.0 m/s to the east.
(a) What is the velocity of the truck right after the collision?
(b) How much mechanical energy is lost in the collision?
To answer these questions, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
The principle of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, assuming no external forces are acting on the system. Mathematically, it can be expressed as:
Total momentum before collision = Total momentum after collision
In equation form, this can be written as:
(mass of car * velocity of car before collision) + (mass of truck * velocity of truck before collision) = (mass of car * velocity of car after collision) + (mass of truck * velocity of truck after collision)
(a) To find the velocity of the truck after the collision, we can rearrange the equation and substitute the given values:
(1225 kg * 25.0 m/s) + (8500 kg * 20.0 m/s) = (1225 kg * 18.0 m/s) + (8500 kg * velocity of truck after collision)
Simplifying the equation:
(30625 kg m/s) + (170000 kg m/s) = (22050 kg m/s) + (8500 kg * velocity of truck after collision)
Subtracting (22050 kg m/s) from both sides:
(30625 kg m/s) + (170000 kg m/s) - (22050 kg m/s) = (8500 kg * velocity of truck after collision)
Simplifying further:
(203575 kg m/s) = (8500 kg * velocity of truck after collision)
Dividing both sides by 8500 kg:
velocity of truck after collision = (203575 kg m/s) / 8500 kg
Therefore, the velocity of the truck right after the collision is approximately 23.97 m/s to the east.
(b) To find the amount of mechanical energy lost in the collision, we need to calculate the initial mechanical energy and the final mechanical energy of the system.
The initial mechanical energy is given by the sum of the kinetic energies of the car and the truck before the collision:
Initial mechanical energy = (0.5 * mass of car * (velocity of car before collision)^2) + (0.5 * mass of truck * (velocity of truck before collision)^2)
Substituting the given values:
Initial mechanical energy = (0.5 * 1225 kg * (25.0 m/s)^2) + (0.5 * 8500 kg * (20.0 m/s)^2)
Simplifying the equation:
Initial mechanical energy = (0.5 * 1225 kg * 625 m^2/s^2) + (0.5 * 8500 kg * 400 m^2/s^2)
Initial mechanical energy = 382812.5 Joules + 1700000 Joules
Initial mechanical energy = 2082812.5 Joules
The final mechanical energy of the system is given by the sum of the kinetic energies of the car and the truck after the collision:
Final mechanical energy = (0.5 * mass of car * (velocity of car after collision)^2) + (0.5 * mass of truck * (velocity of truck after collision)^2)
Substituting the given values:
Final mechanical energy = (0.5 * 1225 kg * (18.0 m/s)^2) + (0.5 * 8500 kg * (23.97 m/s)^2)
Simplifying the equation:
Final mechanical energy = (0.5 * 1225 kg * 324 m^2/s^2) + (0.5 * 8500 kg * 574.0809 m^2/s^2)
Final mechanical energy = 209025 Joules + 2453390.075 Joules
Final mechanical energy = 2662415.075 Joules
Therefore, the amount of mechanical energy lost in the collision is:
Energy lost = Initial mechanical energy - Final mechanical energy
Energy lost = 2082812.5 Joules - 2662415.075 Joules
Energy lost = -579602.575 Joules (negative value indicates energy has been lost)
So, the mechanical energy lost in the collision is approximately 579602.575 Joules.
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of mechanical energy.
(a) Using the principle of conservation of momentum, we can write the equation:
(Mass of car * initial velocity of car) + (Mass of truck * initial velocity of truck) = (Mass of car * final velocity of car) + (Mass of truck * final velocity of truck)
Let's plug in the given values:
(1225 kg * 25.0 m/s) + (8500 kg * 20.0 m/s) = (1225 kg * 18.0 m/s) + (8500 kg * final velocity of truck)
30750 kg·m/s + 170000 kg·m/s = 22050 kg·m/s + (8500 kg * final velocity of truck)
201750 kg·m/s = 22050 kg·m/s + (8500 kg * final velocity of truck)
Now, let's solve for the final velocity of the truck:
201750 kg·m/s - 22050 kg·m/s = 8500 kg * final velocity of truck
179700 kg·m/s = 8500 kg * final velocity of truck
Final velocity of truck = 179700 kg·m/s / 8500 kg
Final velocity of truck ≈ 21.1 m/s to the east
(b) To find the mechanical energy lost in the collision, we can subtract the mechanical energy after the collision from the mechanical energy before the collision. The mechanical energy for an object in motion is given by the equation:
Mechanical energy = 1/2 * (Mass * Velocity^2)
Initial mechanical energy of the system before the collision = (1/2 * 1225 kg * (25.0 m/s)^2) + (1/2 * 8500 kg * (20.0 m/s)^2)
Final mechanical energy of the system after the collision = (1/2 * 1225 kg * (18.0 m/s)^2) + (1/2 * 8500 kg * (21.1 m/s)^2)
Mechanical energy lost = Initial mechanical energy - Final mechanical energy
Calculating the values:
Initial mechanical energy = (1/2 * 1225 kg * (25.0 m/s)^2) + (1/2 * 8500 kg * (20.0 m/s)^2)
Initial mechanical energy ≈ 1,912,500 J
Final mechanical energy = (1/2 * 1225 kg * (18.0 m/s)^2) + (1/2 * 8500 kg * (21.1 m/s)^2)
Final mechanical energy ≈ 1,738,834 J
Mechanical energy lost = 1,912,500 J - 1,738,834 J
Mechanical energy lost ≈ 173,666 J