Law of Gravitation- Find an expression for the magnitude of the initial velocity that a projectile must possess in order to leave the earth when air friction is neglected.

The gravitational constant for g is, G = 6.67428*10^-11 m^3 Kg^-1 s^-2

Please answer the question fully and please give any helpful websites, thankyou.

Also I know calculus.

To find the expression for the magnitude of the initial velocity that a projectile must possess to leave the Earth when neglecting air friction, we need to consider the concept of escape velocity.

The escape velocity is the minimum velocity that an object must have in order to escape the gravitational pull of a planet or any massive body. In the case of leaving the Earth, we can determine the escape velocity using the Law of Gravitation.

The Law of Gravitation states that the force of attraction between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. For a projectile leaving the Earth's surface, the distance can be considered constant.

Let's assume the mass of the Earth is M, the radius of the Earth is R, and the mass of the projectile is m.

From the Law of Gravitation, the force of attraction between the Earth and the projectile can be expressed as:

F = (G * M * m) / R^2

To escape the Earth, the projectile needs to overcome this gravitational force. At the surface of the Earth, the projectile will have an acceleration due to gravity equal to g (approximately 9.8 m/s^2).

Using Newton's second law of motion (F = m * a), we can equate the force of gravity to the force required for the projectile to escape:

(G * M * m) / R^2 = m * g

Simplifying and rearranging the equation:

(G * M) / R^2 = g

Now, we can solve for the escape velocity (v) by equating the kinetic energy of the projectile to the work done against gravity:

(1/2) * m * v^2 = (G * M * m) / R

Canceling out the mass (m) on both sides and rearranging:

v^2 = (2 * G * M) / R

Finally, taking the square root of both sides, we get:

v = √(2 * G * M / R)

Using the given value of the gravitational constant G = 6.67428*10^-11 m^3 Kg^-1 s^-2 and the known values for the mass of the Earth M and the radius of the Earth R, you can calculate the escape velocity from the above expression.

A helpful website that explains the concept of escape velocity and provides further information is:
- https://www.britannica.com/science/escape-velocity

Remember to substitute the appropriate values and units into the equation when performing the calculations.