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.
If fired radically, there is some initial angular momentum, which has to be conserved. At the end, it also has to be conserved, which means it there is some angular KEnergy. So calculate this initial rotational KE, and in the end, it will subtract from the final gravitational PE, you can calculate the distance reached.
Darn it, "If fired tangentally, ..."
Correct the above response.
To calculate the farthest distance the particle reaches when fired tangentially with a speed equal to 3/4 the escape speed, we can use the principle of conservation of mechanical energy.
The escape speed from the surface of a planet can be calculated using the formula:
v_escape = sqrt((2 * G * M) / R)
where v_escape is the escape speed, G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Given that the particle is fired with a speed equal to 3/4 the escape speed, we can calculate the initial kinetic energy (K).
K_initial = (1/2) * m * (3/4 * v_escape)^2
where m is the mass of the particle.
When the particle reaches the farthest distance, its kinetic energy is zero, and its potential energy (U) is maximized.
Since the planet has no atmosphere, the total mechanical energy (E) of the particle remains constant during its trajectory:
E = K_initial + U_max
At the farthest point, the particle's velocity will be perpendicular to the radial direction. Therefore, at that point, its kinetic energy will be zero and its potential energy will be maximized.
Using the principle of conservation of mechanical energy, we have:
E = 0 + U_max
K_initial + U_max = 0 + (-G * M * m / r_max)
where r_max is the farthest distance reached by the particle.
Now, let's plug in the values:
K_initial = (1/2) * m * (3/4 * v_escape)^2
U_max = -G * M * m / r_max
Substituting these values into the conservation of mechanical energy equation, we get:
(1/2) * m * (3/4 * v_escape)^2 - G * M * m / r_max = 0
Simplifying this equation, we have:
(9/32) * v_escape^2 - G * M / r_max = 0
Rearranging the equation to solve for r_max, we get:
r_max = G * M * (32 / 9) / v_escape^2
Substituting the value of v_escape, we have:
r_max = G * M * (32 / 9) / ((2 * G * M) / R)^2
Simplifying further, we obtain:
r_max = (R^2 * 32 / 9) / (4 * G * M / R^2)
Finally, simplifying the expression, we get:
r_max = 8/9 * R
Therefore, the farthest distance the particle reaches, measured from the center of the planet, is 8/9 times the radius (R) of the planet.
To find the farthest distance reached by a particle fired tangentially from the surface of a non-rotating spherical planet, we can use the concept of gravitational potential energy and conservation of mechanical energy.
The escape speed (v_esc) from the surface of a planet can be given by the formula:
v_esc = √(2GM/R)
where G is the universal gravitational constant, M is the mass of the planet, and R is its radius.
In this case, the particle is fired with a speed equal to 3/4 of the escape speed. Let's denote this speed as v_tangential = (3/4)v_esc.
When the particle is fired tangentially, it has both kinetic energy and gravitational potential energy. As it moves away from the planet, its kinetic energy decreases, while its gravitational potential energy increases.
At the farthest distance, the kinetic energy of the particle becomes zero because it comes to a stop. This implies that all of its initial kinetic energy is converted into gravitational potential energy.
The initial kinetic energy (KE_initial) of the particle is given by:
KE_initial = (1/2)mv_tangential^2
where m is the mass of the particle.
The gravitational potential energy (PE_gravitational) of the particle at the farthest distance is given by:
PE_gravitational = G(Mm)/r
where r is the distance from the center of the planet.
Since mechanical energy is conserved, we can equate the initial kinetic energy to the gravitational potential energy:
KE_initial = PE_gravitational
From this equation, we can solve for the farthest distance r:
(1/2)mv_tangential^2 = G(Mm)/r
r = (G(Mm))/[(1/2)mv_tangential^2]
Notice that the mass of the particle 'm' cancels out, and we can substitute v_tangential = (3/4)v_esc. Simplifying further, we get:
r = 2..(G(M))/v_esc^2
Now, you can substitute the given values of M, R, and calculate the escape speed v_esc using the formula mentioned earlier. Once you have the value of v_esc, substitute it into the equation to find the farthest distance reached by the particle fired tangentially from the surface of the spherical planet.