Respuesta :
Answer with explanation:
Relation between Speed , Distance and time
Distance =Speed × Time
→It means Speed is inversely Proportional to time.
As distance will remain constant , in that frame of reference
If speed of light in a Medium
[tex]=s=3 \times 10^8 \frac{\text{meter}}{\text{second}}[/tex]
Then, time taken to cross the medium = t seconds or hours or another unit of time.
Now, If speed of light is 50% of the speed of light relative to us
That is Speed of light in another medium
[tex]=w=\frac{50}{100} \times 3 \times 10^8\\\\=1.5 \times 10^8 \frac{\text{meter}}{\text{second}}[/tex]
Then, time taken to cross the medium = 2t seconds or hours or another unit of time.
Using Unitary Method
[tex]\rightarrow v_{1}\times t_{1}=v_{2} \times t_{2}\\\\1.\rightarrow 3 \times 10^8 \times t=\frac{50}{100} \times 3 \times 10^8 \times t_{1}\\\\t_{1}=2t\\\\2.\rightarrow 3 \times 10^8 \times t=\frac{99.5}{100} \times 3 \times 10^8 \times t_{2}\\\\t_{2}=\frac{1000t}{995}\\\\t_{2}=\frac{200t}{199}[/tex]
Answer:
1.154 times the proper time
10.013 times the proper time
Step-by-step explanation:
Speed of light = c
At 50% speed of light
v = 0.5c
Time dilation
[tex]\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}\\\Rightarrow \Delta t'=\frac{\Delta t}{\sqrt{1-\frac{0.5^2c^2}{c^2}}}\\\Rightarrow \Delta t'=\frac{\Delta t}{\sqrt{1-0.25}}\\\Rightarrow \Delta t'=\frac{\Delta t}{0.866}\\\Rightarrow \Delta t'=1.154\Delta t[/tex]
Time would differ by 1.154 times the proper time
At 99.5% speed of light
v = 0.995c
Time dilation
[tex]\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}\\\Rightarrow \Delta t'=\frac{\Delta t}{\sqrt{1-\frac{0.995^2c^2}{c^2}}}\\\Rightarrow \Delta t'=\frac{\Delta t}{\sqrt{1-0.990025}}\\\Rightarrow \Delta t'=\frac{\Delta t}{0.09987}}\\\Rightarrow \Delta t'=10.013\Delta t[/tex]
Time would differ by 10.013 times the proper time.
