Question

Determine the escape velocity of a projectile fired from the Earth’s Surface, using a differential equation. Compare this with the escape velocity of the other planets, including Pluto.

Answer 1

According to Newton’s law of gravitation, the acceleration of the object will be inversely proportional to the square of the distance from the object to the center of the earth. i.e.,

if a particle at a distance r from the center of the earth [r >R (Radius of Earth)]

Here ,

M be the mass of the Earth,

m be the mass of the projectile,

G is the Universal Gravitational Constant.

We know from Newton's 2nd law that,

    [a=acceleration]

So,

And, the acceleration is given by the differential equation,

and substituting this value we get,

Point to be noted that the acceleration of the object has to be negative because its velocity will be decreasing (the pull of gravity will be slowing it down). So the above equation becomes,

• This is differential equation for the velocity.

Now using the following relation,

we obtain the new form of the above equation that,

Let’s assume that the object leaves the earth’s surface with initial velocity = v₀. Hence, v = v₀ when r = R. From this, we can calculate C:
Using these values we got,

We want to determine the velocity required for the object to escape the gravitational pull of the earth. At r=R, v=v₀
An object projected from the earth with a velocity v₀ such that

Hence, the minimum such velocity, i.e.,

Escape Velocity Equation:

Using the value of g and R we get the value of escape velocity is 11.2.Km/s

*As the expression of the escape velocity contains only r and g so if we can know the value of the other planets, including Pluto.

Here the g and R data should be submitted to get the comparison.