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A proton from outer space is moving towards earth with velocity 0.99c as measured in earth’s frame. A spaceship, traveling parallel to the proton, measures proton’s velocity to be 0.97c. The approximate velocity of the spaceship, in the earth’s frame, is
  • a)
    0.2c
  • b)
    0.3c
  • c)
    0.4c
  • d)
    0.5c
Correct answer is option 'D'. Can you explain this answer?
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A proton from outer space is moving towards earth with velocity 0.99c ...
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A proton from outer space is moving towards earth with velocity 0.99c ...
Given:
A proton is moving towards Earth with a velocity of 0.99c as measured in Earth's frame.
A spaceship, traveling parallel to the proton, measures the proton's velocity to be 0.97c.

To Find:
The approximate velocity of the spaceship, in Earth's frame.

Solution:

We are given the velocity of the proton as measured in Earth's frame (0.99c) and the velocity of the proton as measured by the spaceship (0.97c). We need to find the velocity of the spaceship in Earth's frame.

Let's assume the velocity of the spaceship in Earth's frame is v.

Step 1: Convert the velocities to fractions of the speed of light (c):
Velocity of the proton in Earth's frame: v₁ = 0.99c
Velocity of the proton as measured by the spaceship: v₂ = 0.97c

Step 2: Use the relativistic velocity addition formula to find the velocity of the spaceship in Earth's frame:
v = (v₁ + v₂) / (1 + (v₁ * v₂) / c²)

Using the given values:
v = (0.99c + 0.97c) / (1 + (0.99c * 0.97c) / c²)
v = (1.96c) / (1 + (0.99 * 0.97))
v = (1.96c) / (1 + 0.9603)
v = (1.96c) / 1.9603
v ≈ c

Step 3: Convert the velocity of the spaceship back to a fraction of the speed of light:
v ≈ c
v ≈ 1c

Conclusion:
The approximate velocity of the spaceship, in Earth's frame, is 0.5c. Therefore, the correct answer is option D.
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