### Video Transcript

Suppose a spaceship heading directly away from Earth at 0.750๐ can shoot a
canister at 0.500๐ relative to the ship. What is the velocity of the canister relative to
Earth if it is shot directly at Earth? What is the velocity of the canister relative to Earth
if it is shot directly away from Earth?

Letโs start by highlighting some of the important information given. Weโre told that relative to the Earth. a spaceship is moving away from the Earth
at 0.750๐, where ๐ is the speed of light. And that the spaceship is capable of firing a canister at 0.500๐ in any
direction relative to the ship.

In part one, we want to solve for the velocity of the canister relative to Earth
if it is fired directly at Earth. Weโll call this speed ๐ฃ sub ๐ก for the speed of the canister as it moves toward
Earth. Then we want to solve for the velocity of the canister relative to Earth if itโs
fired directly away from Earth. Weโll call this ๐ฃ sub a.

Letโs draw a diagram of the Earth and the spaceship. Relative to the Earth, the spaceship moves away from the Earth at a speed that
weโll call ๐ฃ sub ๐ . And that speed is 0.750 times the speed of light ๐. Weโre also told that the spaceship is able to fire a canister off at a speed of
0.500๐ relative to the spaceship. Weโll call that ๐ฃ sub ๐. To move towards solving for ๐ฃ sub ๐ก, the speed of the canister as it moves
toward Earth, relative to the Earth, letโs look into a relativistic velocity addition rule thatโs sometimes called
Einstein velocity addition.

If we have two observers A and B where A is considered at rest and B moves
relative to A with a speed weโll call ๐ฃ and if B fires off a projectile that moves away from B with speed ๐ฃ prime, then if we call ๐ข the speed of the projectile relative to the observer at rest
A, then ๐ข is equal to the speed of observer B, ๐ฃ, plus ๐ฃ prime, the speed of the
projectile relative to B, all divided by one plus ๐ฃ times ๐ฃ prime divided by ๐ squared. The plus signs in this equation would change to minus signs if ๐ฃ prime, the
projectileโs velocity, were opposite the direction of ๐ฃ.

So letโs apply this relativistic velocity addition rule to our scenario. We want to solve for ๐ฃ sub ๐ก. Thatโs the velocity of the projectile relative to Earth when the projectile, in our case a canister, is moving directly toward the Earth. That velocity is equal to the speed of the spaceship relative to Earth, ๐ฃ sub ๐ ,
minus ๐ฃ sub ๐ divided by one minus the product ๐ฃ sub ๐ times ๐ฃ sub ๐ over ๐ squared. Weโre using minus signs rather than plus signs because ๐ฃ sub ๐ and ๐ฃ sub ๐ are
in opposite directions from one another. These two speeds are given to us in our problem statement, so we can insert those
values into this relationship.

When we look in the denominator of the equation, we see that ๐, the speed of
light, cancels out, and that the numerator simplifies to 0.250๐. Plugging these values in on our calculator, we find that ๐ฃ sub ๐ก is equal to
0.400 times ๐. Thatโs how fast the canister appears to be moving relative to an observer on
Earth. Notice that our velocity is positive which means that for an observer on Earth,
the canister is still moving away from the Earth.

Now letโs move on to the next part where now the canister is moving away from the Earth in the same direction as
the spaceship. When we apply the relativistic velocity addition rule to this different scenario, solving for ๐ฃ sub a, our equation is the same as it was before except that now instead of minus signs, we have plus signs. And again thatโs because ๐ฃ sub ๐ and ๐ฃ sub ๐ are now acting in the same direction. Once again, we insert our values for these two velocities. We see the factors of ๐, the speed of light, cancel in the denominator, and our numerator sums to 1.250๐.

When we compute this speed, the speed of the canister measured relative to an
observer on Earth when they canister moves away from the Earth, we find a speed of 0.909 times ๐. Notice that this speed is less than the speed of light ๐ even though ๐ฃ sub ๐
plus ๐ฃ sub ๐ is more than the speed of light. This shows our relativistic velocity addition rule in action.