Wavelength Speed Question

[jemjabella42]

Radio waves travel at the speed of light. A satellite is in a "geosynchronous orbit." A radio signal is sent from the ground to the satellite and then the satellite sends the signal back down to the ground. Satellites in geosynchronous orbit are 36,000km above the surface of the earth. How much time does it take for a signal to go from the ground to the satellite and back to the ground?

I plugged numbers into the equation speed=distance/time to get:
2.99792458x10^8 meters per second=36,000km/T

Then, I took both sides multiplied by T to get:
2.99792458x10^8 meters/sec(T)=36,000km

I am not sure how to isolate my variable from here because I'm letting the labels throw me off. I know 1km=1000meters. Do I need to use dimensional analysis to get past this step? It is very confusing for me and I don't know why! It seems like it should be simple math but my brain is just not used to it.

 

[grzz]

Here we go ... like peeling layers off an onion.

v = d/t, where we use 't' for time and reserving 'T' for special intervals of time like the period of a pendulum.

then
3 x 10^8 = 2 x 36 x10^6/t thus converting km into m, noting that the signal has to return back.

Now let us 'peel off' t from the RHS

3 x 10^8 x t = 2 x 36 x10^6 .....multiplying both sides by t

then we 'peel off 3 x 10^8 from the LHS

t = 2 x 36 x10^6/3 x 10^8.....dividing both sides by 3 x 10^8

t = ...

 

[jemjabella42]

Here we go ... like peeling layers off an onion.

v = d/t, where we use 't' for time and reserving 'T' for special intervals of time like the period of a pendulum.

then
3 x 10^8 = 2 x 36 x10^6/t thus converting km into m, noting that the signal has to return back.

Now let us 'peel off' t from the RHS

3 x 10^8 x t = 2 x 36 x10^6 .....multiplying both sides by t

then we 'peel off 3 x 10^8 from the LHS

t = 2 x 36 x10^6/3 x 10^8.....dividing both sides by 3 x 10^8

t = ...

I get confused when converting. I understand now why I need to multiply by 2 but how exactly do I arrive at 36x10^6? Could you explain this part in more detail please?

 

[HallsofIvy]

What do you mean by "arrive at 36x10^6"? It is given in the problem that the satellite is 36000 km above the earth. Presumably you know that there are 1000 meters in a kilometer (if you don't then it is worth memorizing "kilo= 1000") so it is 36000*(1000)= 36000000= 36*10000000= 36*10^6 meters above the earth.

 

[asteriatic]

As a scientist, it is important to remember that units are just labels and can be converted as needed. In this scenario, we can use dimensional analysis to convert the distance of 36,000km to meters. We know that 1km is equal to 1000m, so we can multiply 36,000km by 1000m/1km to get 36,000,000m.

Now, we can plug this value into our equation: 2.99792458x10^8 meters/sec(T)=36,000,000m. To isolate the variable T, we can divide both sides by 2.99792458x10^8 meters/sec, giving us T=36,000,000m/2.99792458x10^8 meters/sec.

Simplifying this further, we get T=0.12 seconds. Therefore, it takes approximately 0.12 seconds for a radio signal to travel from the ground to a satellite in geosynchronous orbit and back to the ground. This calculation highlights the incredible speed at which radio waves travel and the efficiency of using satellites in geosynchronous orbit for communication purposes.