Derive Relation between orbital velocity and escape velocity
Answers
The square of the orbital velocity is equal to the centripetal acceleration multiplied by the radius of the circular path. In symbols v^2 = a * R.
By replacing a by g and R is the radius of the earth from its surface the formula becomes the orbital velocity of a satellite v^2 = g * R.
g = 9.81 m/s^2 and R = the radius of the earth which is 6.37 x 10^6 meters.
To escape the earth’s gravity, the formula becomes v^2 = 2 * g * R. This was derived from K.E. = P. E conservation of energy formula.
1/2 * m * v^2 = m * g * h and becomes 1/2 * v^2 = g * h and v^2 = 2 * g * h and by replacing h by R measured from the earth’s surface v^2 = 2 * g * R where R = 6.37 x 10^6 meters and g = 9.81 m/s^2.
The required speed is about 11 km/s. Why is this the formula? Well it was derived and tested so many times and found to be always true of all escaping masses. The mass of the body is not even included.
If you go to the moon the g and R must be that of the moon. If you want to escape the planet Jupiter you must use the g and R of Jupiter.
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here's your answer:
As, we know that the force of gravity brings back bodies thrown upwards if their initial velocities is less than the escape velocity. The velocity allowing a body to move in circular path around the earth is called its orbital velocity.
The square of the orbital velocity is equal to the centripetal acceleration multiplied by the radius of the circular path. In symbols v^2 = a * R.
By replacing a by g and R is the radius of the earth from its surface the formula becomes the orbital velocity of a satellite v^2 = g * R.
g = 9.81 m/s^2 and R = the radius of the earth which is 6.37 x 10^6 meters.
To escape the earth’s gravity, the formula becomes v^2 = 2 * g * R. This was derived from K.E. = P. E conservation of energy formula.
1/2 * m * v^2 = m * g * h and becomes 1/2 * v^2 = g * h and v^2 = 2 * g * h and by replacing h by R measured from the earth’s surface v^2 = 2 * g * R where R = 6.37 x 10^6 meters and g = 9.81 m/s^2.
The required speed is about 11 km/s. Why is this the formula? Well it was derived and tested so many times and found to be always true of all escaping masses. The mass of the body is not even included.
If you go to the moon the g and R must be that of the moon. If you want to escape the planet Jupiter you must use the g and R of Jupiter.
i hope it will helps you