Hi. I am Walter White. I am good at chemistry but suck at physics. Can you help me?
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Bob is traveling in a very fast space (v=0.8c) away from Alice, who is sitting still in an observatory watching him through a very good telescope.
Bob is throwing stuff off the ship. He first throws a garbage bag out of the bow hatch, waits 20 seconds and then tosses another one from the stern hatch which is 20m away.
Meanwhile, Alice tosses garbage at a basket near her and misses. The garbage basket is 5m away and it takes 2.5s for it to reach (but miss) the basket.
a) According to Alice, how long a time interval passes between the first and second garbage bag through by Bob?
b) According to Bob, how long a time interval passes between Alice throwing her garbage and it missing the basket.
c) According to Alice, how far apart are the two hatches on Bob's ship?
d) According to Bob, how far apart are Alice and the garbage basket?
Special relativity questions. Thank you in advanced.
Hi Walter White! I'd be happy to help you with your physics questions. Let's go through each question step by step and explain how to solve them.
a) According to Alice, how long a time interval passes between the first and second garbage bag thrown by Bob?
To solve this, we need to use the concept of time dilation in special relativity. Since Bob is traveling at a speed of 0.8c (80% of the speed of light), time will appear to pass slower for him relative to Alice, who is sitting still in the observatory.
We know that Bob throws the second garbage bag 20 seconds after throwing the first one. However, from Alice's perspective, she will see time dilated. The time dilation factor is given by the Lorentz transformation:
γ = 1 / √(1 - v^2/c^2)
where c is the speed of light in a vacuum (3 x 10^8 m/s) and v is Bob's velocity relative to Alice.
In this case, v = 0.8c, so we can substitute these values into the equation and calculate the time dilation factor γ.
Once we have γ, we can multiply it by the 20 seconds that Alice observes and find the time interval as seen by Alice.
b) According to Bob, how long a time interval passes between Alice throwing her garbage and it missing the basket?
Again, we need to account for time dilation here, but this time from Bob's perspective. Bob is moving away from Alice, so he will perceive time to be passing slower for Alice compared to himself.
We know that it takes 2.5 seconds for Alice's garbage to reach the basket, but it misses. To find the time interval as seen by Bob, we need to divide this time by the time dilation factor γ.
We already found γ in the previous question, so we can use it to calculate the time interval as perceived by Bob.
c) According to Alice, how far apart are the two hatches on Bob's ship?
To determine the distance between the two hatches on Bob's ship as seen by Alice, we need to account for the effects of length contraction in special relativity. When an object is moving relative to an observer, its length in the direction of motion appears shorter.
To calculate the contracted length, we can use the formula:
L' = L * √(1 - v^2/c^2)
where L is the rest length (the distance between the hatches when the ship is at rest) and v is Bob's velocity.
We know that the distance between the hatches when the ship is at rest is 20 meters, so we can substitute this value along with Bob's velocity (0.8c) into the formula to find the contracted length as seen by Alice.
d) According to Bob, how far apart are Alice and the garbage basket?
Similarly, to determine the distance between Alice and the garbage basket as seen by Bob, we need to account for length contraction. Bob is observing Alice and the garbage basket moving away from him at 0.8c.
We know that the distance between Alice and the basket is 5 meters, so we can use the length contraction formula with v = 0.8c and L = 5 meters to calculate the contracted length as seen by Bob.
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To summarize, we need to:
a) Calculate the time interval between the two garbage bags thrown by Bob as seen by Alice, using time dilation.
b) Calculate the time interval between Alice throwing her garbage and it missing the basket as seen by Bob, using time dilation.
c) Calculate the distance between the two hatches on Bob's ship as seen by Alice, using length contraction.
d) Calculate the distance between Alice and the garbage basket as seen by Bob, using length contraction.
I hope this provides you with a clear explanation of how to solve these problems!