A 2,000 kg car moving at 10 m/s collides head-on with a 2,500 kg car moving in the opposite direction at 15 m/s. The two cars are locked together after impact.
Is this an elastic or an inelastic collision? Why?
What is the speed of the cars after impact?
Calculate the kinetic energies of the cars, both before and after impact.
What fraction of the kinetic energy was lost during the impact? Where did the energy go?
idk
This is an inelastic collision because the two cars stick together after the impact. In inelastic collisions, kinetic energy is not conserved and is converted into other forms of energy, such as deformation or heat.
To find the speed of the cars after impact, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Let's define the positive direction as the direction of the first car's initial velocity:
The momentum before the collision is given by:
Initial momentum of car 1 = (mass of car 1) * (velocity of car 1)
= (2000 kg) * (10 m/s)
Initial momentum of car 1 = 20,000 kg m/s
The momentum of car 2 before the collision is:
Initial momentum of car 2 = (mass of car 2) * (velocity of car 2)
= (2500 kg) * (-15 m/s) (negative because it's moving in the opposite direction)
Initial momentum of car 2 = -37,500 kg m/s
The total initial momentum is the sum of the individual momenta:
Total initial momentum = (Initial momentum of car 1) + (Initial momentum of car 2)
= 20,000 kg m/s + (-37,500 kg m/s)
Total initial momentum = -17,500 kg m/s
After the collision, the two cars stick together, so they will have a combined mass of 2000 kg + 2500 kg = 4500 kg.
To find the final velocity of the combined cars, we divide the total initial momentum by the combined mass:
Final velocity of combined cars = (Total initial momentum) / (Combined mass)
= (-17,500 kg m/s) / (4500 kg)
Final velocity of combined cars ≈ -3.89 m/s
The speed of the cars after impact is approximately 3.89 m/s in the opposite direction.
Now let's calculate the kinetic energies:
The kinetic energy before the collision is given by:
Initial kinetic energy of car 1 = 1/2 * (mass of car 1) * (velocity of car 1)^2
= 1/2 * (2000 kg) * (10 m/s)^2
Initial kinetic energy of car 1 = 100,000 J
The kinetic energy before the collision of car 2 is:
Initial kinetic energy of car 2 = 1/2 * (mass of car 2) * (velocity of car 2)^2
= 1/2 * (2500 kg) * (15 m/s)^2
Initial kinetic energy of car 2 = 281,250 J
The total initial kinetic energy is the sum of the individual kinetic energies:
Total initial kinetic energy = (Initial kinetic energy of car 1) + (Initial kinetic energy of car 2)
= 100,000 J + 281,250 J
Total initial kinetic energy = 381,250 J
The kinetic energy after the collision is given by:
Final kinetic energy = 1/2 * (total mass after collision) * (final velocity of combined cars)^2
= 1/2 * (4500 kg) * (-3.89 m/s)^2
Final kinetic energy ≈ 35,501 J
To find the fraction of the kinetic energy lost during the impact, we compare the difference in kinetic energy before and after the collision:
Kinetic energy lost = (Total initial kinetic energy) - (Final kinetic energy)
= 381,250 J - 35,501 J
Kinetic energy lost ≈ 345,749 J
The fraction of the kinetic energy lost during the impact is given by:
Fraction of kinetic energy lost = (Kinetic energy lost) / (Total initial kinetic energy)
= 345,749 J / 381,250 J
Fraction of kinetic energy lost ≈ 0.91
During the impact, the energy lost is converted into other forms, such as deformation of the cars and heat.
To determine whether the collision is elastic or inelastic, we need to consider whether kinetic energy is conserved or not. In an elastic collision, both momentum and kinetic energy are conserved, while in an inelastic collision, only momentum is conserved.
First, let's calculate the total momentum of the system before and after the collision. The total momentum is the sum of the individual momenta of the two cars.
Before the collision:
Car 1 momentum = mass (m1) × velocity (v1) = 2,000 kg × 10 m/s = 20,000 kg⋅m/s (since it's moving in one direction)
Car 2 momentum = mass (m2) × velocity (v2) = 2,500 kg × (-15 m/s) = -37,500 kg⋅m/s (since it's moving in the opposite direction)
Total momentum before collision: 20,000 kg⋅m/s - 37,500 kg⋅m/s = -17,500 kg⋅m/s (the negative sign indicates the opposite direction)
After the collision, the two cars are locked together, so they move as one unit. The total momentum of the system after the collision is equal to the mass of the combined cars (m1 + m2) times their final velocity (vf).
Since the two cars are locked together and both have the same final velocity, let's call this final velocity vf.
After the collision:
Total momentum after collision = (m1 + m2) × vf
Applying the principle of conservation of momentum:
Total momentum before collision = Total momentum after collision
-17,500 kg⋅m/s = (2,000 kg + 2,500 kg) × vf
-17,500 kg⋅m/s = 4,500 kg × vf
Solving for vf:
vf = -17,500 kg⋅m/s / 4,500 kg
vf ≈ -3.89 m/s
Since the velocity is negative, it means the cars are moving in the opposite direction after the collision as well.
Now, let's calculate the kinetic energy before and after the collision.
The kinetic energy (KE) of an object is given by the formula: KE = (1/2) × mass × velocity^2.
For the first car (car 1):
KE1 = (1/2) × 2,000 kg × (10 m/s)^2 = 100,000 J
For the second car (car 2):
KE2 = (1/2) × 2,500 kg × (15 m/s)^2 = 281,250 J
Total kinetic energy before collision: KE1 + KE2 = 100,000 J + 281,250 J = 381,250 J
After the collision, the two cars move together as one unit, so their combined kinetic energy is:
KEf = (1/2) × (m1 + m2) × vf^2 = (1/2) × 4,500 kg × (-3.89 m/s)^2
KEf ≈ 29,730 J
Now, let's calculate the fraction of the kinetic energy lost during the impact:
Kinetic energy lost = Initial kinetic energy - Final kinetic energy
Kinetic energy lost = 381,250 J - 29,730 J = 351,520 J
Fraction of kinetic energy lost = Kinetic energy lost / Initial kinetic energy
Fraction of kinetic energy lost = 351,520 J / 381,250 J ≈ 0.922 or 92.2%
So, approximately 92.2% of the kinetic energy was lost during the impact. The energy is usually lost in the form of heat, sound, deformation of the cars, and other mechanical work done during the collision.