Hi, can you please help me answer these following questions?
1. You have won another space trip. This time you will be traveling at a constant speed of`2.90*10^8` m/s relative to earth. Now your best friend had to stay behind on Earth. Your friend aged 25 years during the time of your trip. How many years did you age?
2.Now your friend is traveling at a velocity of 0.80 c towards you. You are stationary on Earth. Your friend throws a baseball forward at a velocity of 0.60 c(relative to your friend). What is the velocity of the baseball with respect to you?
3. The mass of beryllium nucleus (7, 4 Be) is 1.1652 x 10^-26kg. What is the mass defect of the beryllium nucleus? What is the binding energy of the beryllium nucleus? What is the binding energy per nucleon on the beryllium nucleus?
1. You need to use the relativistic time dialation formula:
`t = t_0/sqrt(1 - v^2/c^2)` , where `t_0` is the time elapsed in Earth reference system, 25 years, v is the speed you have traveled with, `2.9 * 10^8 m/s` , and c is the speed of light, `3* 10*8 m/s` .
Thus, the time that elapsed in your reference system, and this is how much you aged, is
`t = 25/sqrt(1 - 2.9/3)=136.9` years
The answer is 136.9 years.
2. You need to use the formula for the addition of relativistic speeds:
`v=(v_0+u)/(1+(v_0*u)/c^2)` . where u is the speed of the reference system that is moving (your friend), `v_0` is the speed of the baseball in the moving system (your friend) and v is the speed of the baseball in the stationary system (you). And, c is again the speed of light.
Plugging in u = .8c and `v_0 = .6c` you get
The velocity of the baseball with respect to you is `35/37 c` .
2) For this question, you need the formula for the relativistic addition of velocities:
u = (v + u')/(1 + vu'/c^2)
Here, v is the velocity of your friend relative to you, u' is the velocity of the ball relative to your friend, and u is the velocity of the ball relative to you.
If v = 0.8c and u' = 0.6c, then
u = (0.8c + 0.6c)/(1 + (0.8c*0.6c)/c^2) = 1.4c/(1 + 0.48) = 0.95c
The speed of the baseball with respect to you is 0.95c.
I am going to answer the first two questions.
1) If you are traveling with the speed close to the speed of light (`c=3*10^8 m/s)` , the clocks in your reference frame will be running slow compared to the stationary clocks on Earth. So, if you friend aged 25 years, you will have aged the number of years less by the factor of
`gamma = 1/sqrt(1-v^2/c^2)` (Here, v is your speed relative to Earth.)
If you are traveling at a speed of `2.9*10^8` m/s, then
`v/c = (2.9*10^8)/(3*10^8) = 0.967`
`gamma = 1/sqrt(1-(0.967)^2) = 3.91`
and you will have aged by 25/3.91 = 6.4 years.