Take into account the Earth's rotational speed (1 rev/day) and determine the necessary speed, with respect to the Earth, for a rocket to escape if fired from the Earth at the equator in a direction a) eastward; b) westward; c) vertically upward.
The answers are 1.07 x10^4 m/s, 1.17 x10^4 m/s and 1.12 x10^4 m/s. I got c but I don't understand what to do for a and b.
The sun rises in the East.
Therefore even if stationary, you are already moving East with the surface of the earth. That Eastward speed is 2 pi*earth radius / (24 hr*3600 seconds/hr)
If you launch Eastward, your total speed is the speed relative to earth PLUS the speed of that point on the earth. If you launch westward your total speed is the speed relative to earth MINUS the speed of that point on the earth. Therefore you will need more rocket push to reach escape speed if you launch westward.
To determine the necessary speed for a rocket to escape Earth when fired from the equator, we need to consider the Earth's rotational speed and its effect on the rocket's velocity.
a) Eastward:
When a rocket is fired in the eastward direction (parallel to the Earth's rotation), it receives an additional boost from the Earth's rotational velocity. To calculate the necessary speed for escape, we need to find the total velocity required, which consists of two components: the Earth's rotational velocity and the escape velocity.
The Earth's rotational speed is approximately 1 revolution per day or 360 degrees per 24 hours. To convert this into meters per second, we need to determine how far the equator travels in one second. We can use the circumference of the Earth at the equator, which is approximately 40,075 km.
1 revolution = 360 degrees = 40,075 km
1 degree = 40,075 km / 360
1 degree = 111.32 km
1 degree = 111,320 m
To determine the necessary speed for escape, we add the Earth's rotational velocity to the escape velocity. The escape velocity from Earth is approximately 11.2 km/s (or 11,200 m/s).
Necessary speed for escape = Earth rotational speed + Escape velocity
Necessary speed for escape = (111,320 m/second) + (11,200 m/second)
Necessary speed for escape = 122,520 m/second
Therefore, the necessary speed, with respect to the Earth, for a rocket to escape when fired eastward from the equator is approximately 1.23 x 10^5 m/s (not 1.07 x 10^4 m/s).
b) Westward:
Similarly, when a rocket is fired westward (opposite to the Earth's rotation), it needs to overcome the Earth's rotational velocity.
Using the same calculations as in a), we find that the total velocity required for escape westward is:
Necessary speed for escape = Earth rotational speed - Escape velocity
Necessary speed for escape = (111,320 m/second) - (11,200 m/second)
Necessary speed for escape = 100,120 m/second
Therefore, the necessary speed, with respect to the Earth, for a rocket to escape when fired westward from the equator is approximately 1.00 x 10^5 m/s (not 1.17 x 10^4 m/s).
c) Vertically upward:
When a rocket is fired vertically upward, it does not gain or lose any additional velocity from the Earth's rotation. Therefore, we only need to consider the escape velocity.
Using the escape velocity of 11.2 km/s (or 11,200 m/s), the necessary speed for the rocket to escape vertically upward is:
Necessary speed for escape = Escape velocity
Necessary speed for escape = 11,200 m/second
Therefore, the necessary speed, with respect to the Earth, for a rocket to escape when fired vertically upward from the equator is approximately 1.12 x 10^4 m/s, which matches the provided answer.