Take into account the Earth's rotational speed (1 rev/day) and use 5 significant figures.
(a) Determine the necessary speed, with respect to Earth, for a rocket to escape if fired from the Earth at the equator in a direction eastward. (The escape velocity from the surface of the Earth is 11.182 km/s. The radius of the Earth is 6380 km.)
(b) Determine the necessary speed, with respect to Earth, for a rocket to escape if fired in a westward direction at the equator.
(c) Determine the necessary speed, with respect to Earth, for a rocket to escape if fired in a westward direction at 20° north latitude.
(d) Determine the necessary speed, with respect to Earth, for a rocket to escape if fired from the Earth at the equator vertically upward.
To determine the necessary speed for a rocket to escape Earth in different scenarios, we need to consider the Earth's rotational speed and the escape velocity.
The escape velocity is the minimum speed an object must reach to overcome the gravitational pull of a planet and escape its gravitational field. For the Earth, the escape velocity from the surface is 11.182 km/s.
(a) To determine the necessary speed for a rocket fired from the Earth at the equator in an eastward direction, we need to account for the Earth's rotational speed. The Earth completes one full rotation in approximately 24 hours or 86400 seconds. This means that at the equator, the Earth's rotational speed is approximately 460 m/s (rounded to three significant figures).
To calculate the necessary speed, we need to add the Earth's rotational speed to the escape velocity:
Necessary speed = Escape velocity + Rotational speed.
Necessary speed = 11.182 km/s + 0.460 km/s = 11.642 km/s (rounded to five significant figures).
(b) If the rocket is fired in a westward direction at the equator, we still need to account for the Earth's rotational speed. Since the direction is opposite to the Earth's rotation, we subtract the rotational speed from the escape velocity:
Necessary speed = Escape velocity - Rotational speed.
Necessary speed = 11.182 km/s - 0.460 km/s = 10.722 km/s (rounded to five significant figures).
(c) When the rocket is fired in a westward direction at a latitude of 20° north, we need to consider the effect of both the Earth's rotational speed and the latitude. At this latitude, the rotational speed is reduced by a factor equal to the cosine of the latitude.
First, we calculate the reduced rotational speed:
Reduced rotational speed = Rotational speed * cos(latitude).
Reduced rotational speed = 0.460 km/s * cos(20°) = 0.431 km/s (rounded to three significant figures).
Then, we calculate the necessary speed by subtracting the reduced rotational speed from the escape velocity:
Necessary speed = Escape velocity - Reduced rotational speed.
Necessary speed = 11.182 km/s - 0.431 km/s = 10.751 km/s (rounded to five significant figures).
(d) If the rocket is fired vertically upward from the equator, the Earth's rotational speed is perpendicular to the rocket's motion and does not affect the escape velocity. Therefore, the necessary speed is equal to the escape velocity:
Necessary speed = Escape velocity = 11.182 km/s (rounded to five significant figures).
To summarize:
(a) Necessary speed (eastward) = 11.642 km/s.
(b) Necessary speed (westward) = 10.722 km/s.
(c) Necessary speed (westward, 20° N latitude) = 10.751 km/s.
(d) Necessary speed (vertical, equator) = 11.182 km/s.