Projectile moving from Earth's surface

[JasonV]

A projectile is shot directley away from Earth's surface. Neglect the rotation of Earth. What multiple of Earth's radius RE gives the radial distance a projectile reaches if (a) its initial is .500 of the escape speed from Earth and (b) its initial kinetic energy is .500 of the kinetic energy required to escape Earth? (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth?

I think i got (a) and (b) but i need help on (c).
I don't understand how to complete the problem without knowing the mass of the projectile.

Please help.

 

[Dick]

If you got a) and b) then you don't need much help from me. Just express c) in terms of m. Where you don't know m. Leave it as a variable.

 

[ogb p]

I would like to clarify that the initial kinetic energy and escape speed of a projectile are dependent on the mass of the object. Therefore, in order to accurately answer parts (a) and (b), we would need to know the mass of the projectile. Additionally, the least initial mechanical energy required for the projectile to escape Earth would also depend on its mass.

To solve for the radial distance a projectile reaches in part (a), we can use the equation for escape speed:

v = √(2GM/R)

Where v is the escape speed, G is the gravitational constant, M is the mass of Earth, and R is the radius of Earth.

Since the initial velocity is 0.5 of the escape speed, we can set up the equation:

0.5v = √(2GM/R)

Solving for R, we get:

R = (GM)/(0.5v)^2

Similarly, for part (b), we can use the equation for kinetic energy:

KE = (1/2)mv^2

Where KE is the kinetic energy, m is the mass of the projectile, and v is the initial velocity.

Since the initial kinetic energy is 0.5 of the energy required to escape, we can set up the equation:

0.5KE = (1/2)m(0.5v)^2

Solving for m, we get:

m = (2KE)/v^2

Therefore, the radial distance reached in part (b) can be calculated using the same equation as in part (a), but with the calculated mass instead.

For part (c), the least initial mechanical energy required for the projectile to escape Earth would be the sum of its initial kinetic energy and its initial potential energy. Since the initial potential energy is 0 (since we are neglecting the rotation of Earth), the least initial mechanical energy required would simply be the initial kinetic energy, which we calculated in part (b).

In conclusion, without knowing the mass of the projectile, we cannot accurately solve for parts (a) and (b). And the least initial mechanical energy required for the projectile to escape Earth would simply be its initial kinetic energy, which would depend on its mass.