A non-rotating spherical planet with no atmosphere has a mass M and radius R. A particle is fired off from the surface with a speed equal to 3/4 the escape speed. Calculate the farthest distance it reaches (measured from the center of the planet) if it is fired tangentially. I know how solve it if fired radially, but I don't know how to do it if the particle is fired tangentially.
To solve this problem, we need to use the concept of kinetic and potential energy to find the maximum distance the particle reaches when fired tangentially from the surface of the planet.
First, let's recall the formula for escape velocity (Ve) from the surface of a planet:
Ve = √(2GM/R)
where G is the universal gravitational constant, M is the mass of the planet, and R is its radius.
Given that the particle is fired with a speed of 3/4 the escape velocity, the initial speed (V0) of the particle is:
V0 = (3/4)Ve
To find the maximum distance (d) the particle reaches, we need to equate the initial kinetic energy with the final potential energy. This occurs when the particle reaches its maximum height and momentarily comes to rest.
The initial kinetic energy (K0) of the particle is given by:
K0 = (1/2)mV0^2
where m is the mass of the particle.
The final potential energy (Pf) of the particle at its maximum altitude is given by:
Pf = -GMm/d
Equating K0 and Pf, we have:
(1/2)mV0^2 = -GMm/d
Simplifying the equation, we get:
d = -(GMV0^2)/(2K0)
Substituting the values, we have:
d = -(GM(3/4)^2Ve^2)/(2(1/2)m(3/4)^2Ve^2)
Simplifying further:
d = -(GMVe^2)/[(8m/3)(Ve^2)]
d = -(3GM)/(8m)
Therefore, the farthest distance (d) the particle reaches when fired tangentially from the surface is -(3GM)/(8m). Please note that the negative sign indicates that the distance is measured from the center of the planet, and the result is proportional to the mass and inversely proportional to the mass of the particle.