A lighthouse sweeps its beam of light around in a circle once every 7.2 s. To an observer in a spaceship moving away from Earth, the beam of light completes one full circle every 9 s. What is the speed of the spaceship relative to Earth
To find the speed of the spaceship relative to Earth, we can first calculate the angular speed of the lighthouse's beam of light.
The angular speed is defined as the angle covered per unit time. In this case, the angle covered is one full circle, which is 360 degrees or 2π radians, and the time taken is 7.2 seconds. So the angular speed (ω) of the lighthouse's beam is:
ω = (angle covered) / (time taken) = 2π radians / 7.2 seconds
Next, we can find the angular speed of the lighthouse's beam as observed by the observer in the spaceship. The beam completes one full circle every 9 seconds, so the angular speed (ω') as observed by the observer is:
ω' = (angle covered) / (time taken) = 2π radians / 9 seconds
The angular speed seen by the observer will be different due to the relative motion between the spaceship and Earth.
Now, we can use the formula for relative angular speed to find the speed of the spaceship relative to Earth. The relative angular speed is given by:
ω' - ω = (speed of spaceship) / (distance of observer from lighthouse)
Since the distance of the observer from the lighthouse is not mentioned, we can assume it remains constant.
Rearranging the formula, we can solve for the speed of the spaceship:
(speed of spaceship) = (ω' - ω) * (distance of observer from lighthouse)
Therefore, to find the speed of the spaceship relative to Earth, we need to know the value of ω', ω, and the distance of the observer from the lighthouse. With this information, we can plug it into the formula and calculate the speed of the spaceship.