Question
upstudy study bank question image url

Observer on earth sees a ship travelling at \( 0.5 \mathrm{c}[\mathrm{fwd}] \) wrt the ground. A plasma cannon fires a pulse out the front of the ship so that the ground observer sees the pulse travel at 0.9 c [fwd] wrt the ground. How fast do the observers in the ship see the pulse travel wrt the ship?

Ask by Chandler Sullivan. in Canada
Mar 09,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The speed of the pulse with respect to the ship is approximately 0.9655 c.

Solution

Let's denote the speed of the pulse as \( v \) and the speed of the ship as \( v_{\text{ship}} \). Given that the observer on earth sees the ship traveling at \( 0.5c \) and the pulse traveling at \( 0.9c \) with respect to the ground, we can use the relativistic velocity addition formula to find the speed of the pulse with respect to the ship. The relativistic velocity addition formula is given by: \[ v_{\text{pulse}} = \frac{v + v_{\text{ship}}}{1 + \frac{v \cdot v_{\text{ship}}}{c^2}} \] Substitute the given values into the formula: \[ v_{\text{pulse}} = \frac{0.9c + 0.5c}{1 + \frac{0.9c \cdot 0.5c}{c^2}} \] Simplify the expression to find the speed of the pulse with respect to the ship. Simplify the expression by following steps: - step0: Solution: \(\frac{\left(0.9c+0.5c\right)}{\left(1+\frac{\left(0.9c\times 0.5c\right)}{c^{2}}\right)}\) - step1: Remove the parentheses: \(\frac{0.9c+0.5c}{1+\frac{0.9c\times 0.5c}{c^{2}}}\) - step2: Reduce the fraction: \(\frac{0.9c+0.5c}{1+\frac{9}{20}}\) - step3: Add the terms: \(\frac{1.4c}{1+\frac{9}{20}}\) - step4: Add the numbers: \(\frac{1.4c}{\frac{29}{20}}\) - step5: Rewrite the expression: \(\frac{\frac{7}{5}c}{\frac{29}{20}}\) - step6: Rewrite the expression: \(\frac{\frac{7c}{5}}{\frac{29}{20}}\) - step7: Multiply by the reciprocal: \(\frac{7c}{5}\times \frac{20}{29}\) - step8: Reduce the fraction: \(7c\times \frac{4}{29}\) - step9: Multiply the terms: \(\frac{7c\times 4}{29}\) - step10: Multiply the terms: \(\frac{28c}{29}\) The speed of the pulse with respect to the ship is \( \frac{28c}{29} \) or approximately 0.9655c.

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Bonus Knowledge

To find the speed of the pulse as observed from the ship, we can use the relativistic velocity addition formula. This formula states that if an object (like the pulse) moves at velocity \( u' \) in a frame moving at velocity \( v \), the velocity \( u \) in the stationary frame is given by: \[ u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \] In this scenario: - The ship's speed \( v = 0.5c \) - The pulse speed relative to the ground \( u = 0.9c \) We need to find \( u' \) (the speed of the pulse relative to the ship). Using the rearranged formula: \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \] Substituting the values: \[ u' = \frac{0.9c - 0.5c}{1 - \frac{(0.9c)(0.5c)}{c^2}} = \frac{0.4c}{1 - 0.45} = \frac{0.4c}{0.55} \approx 0.727c \] Thus, the observers in the ship see the pulse travel at approximately \( 0.727c \) with respect to the ship.

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy