A 2.4 Kg object moving to the right at 0.8c collides with a 3.2 kg object moving to left at 0.6c. They stick
together. Find the mass and velocity of the resulting combined object by;
a) assuming that you have no knowledge of relativistic physics,
b) using the special theory of relativity.
c) Compute the chance in the kinetic energy, what does it correspond to?
Well, we knoew that M1 = 2.4kg, and m2 = 3.2kg. We can then use the convservation of momentum law to find the resulting velocity and momentum after the collision. Since they stick together, we can add their masses.
m1v1 + m2v2 = (m1+m2)v
so
(2.4)(.8) + (3.2)(.6) = (2.4+3.2)v
1.92 + (-1.92) = 5.6v
0 = 5.6v
v = 0
Therefore, the carts stick together and stop.
CHANGE IN KINETIC ENERGY:
EK2 - EK1
1/2mv^2 - 1/2mv^2
1/2(2.4)(.8)^2 - 1/2(3.2)(.6)^2
.768 - .576
= .192J
To find the mass and velocity of the resulting combined object, we can approach it using both classical mechanics (assuming no knowledge of relativistic physics) and the special theory of relativity.
a) Assuming no knowledge of relativistic physics (classical mechanics):
In classical mechanics, we can apply the principle of conservation of momentum. The sum of momentum before the collision will be equal to the sum of momentum after the collision.
1. Find the initial momentum:
The momentum of an object can be calculated by multiplying its mass with its velocity. In this case:
Object 1 (moving to the right):
Momentum1 = mass1 * velocity1 = 2.4 kg * (0.8c)
Object 2 (moving to the left):
Momentum2 = mass2 * velocity2 = 3.2 kg * (-0.6c) [Note: Velocity to the left has a negative sign]
2. Find the total momentum before the collision:
Total Initial Momentum = Momentum1 + Momentum2
3. Find the mass and velocity of the resulting combined object:
Total momentum after the collision will be equal to the momentum of the combined object.
Momentum of the combined object = Total Initial Momentum
Mass of the combined object = mass1 + mass2
Velocity of the combined object = Total Initial Momentum / (mass1 + mass2)
b) Using the special theory of relativity:
In relativistic physics, we need to consider the relativistic addition of velocities and the mass-energy equivalence principle.
1. Find the initial energy and momentum:
The energy of an object can be calculated using Einstein's mass-energy equivalence principle: E = mc^2. Remember to convert the mass from kg to joules using the equation: E = mc^2.
Object 1:
Energy1 = mass1 * c^2 [where c is the speed of light]
Momentum1 = Energy1 / (velocity1 * c)
Object 2:
Energy2 = mass2 * c^2
Momentum2 = Energy2 / (velocity2 * c)
2. Find the total energy and momentum before the collision:
Total Initial Energy = Energy1 + Energy2
Total Initial Momentum = Momentum1 + Momentum2
3. Find the mass and velocity of the resulting combined object:
Using the relativistic addition of velocities formula, we can find the velocity of the combined object:
Velocity of the combined object = (Momentum1 + Momentum2) / (Energy1 + Energy2)
To find the mass, we need to use the mass-energy equivalence principle:
Mass of the combined object = (Total Initial Energy) / c^2
c) To compute the change in kinetic energy and what it corresponds to:
The change in kinetic energy can be computed by subtracting the initial kinetic energy of the two objects from the final kinetic energy of the combined object.
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
The change in kinetic energy corresponds to the energy lost or gained during the collision. Positive change indicates energy gained, and negative change indicates energy lost.
Please note that in relativistic physics, the equations mentioned above are simplified versions for low velocities and small masses compared to the speed of light. For high velocities or significant masses, the full equations of special relativity should be used.