Gravitational Slingshot

Question

A satellite of mass m_s and speed v_s,i circles around a planet of mass m_p that is moving with speed v_p,i in the opposite direction.

Since gravity is a conservative force, the collision is elastic.

m_p = 2.20E+24 kg

m_s = 1460 kg

v_s,i  x = 1.748E+4 m/s

v_p,i  x = -9.20E+3 m/s

a) find the final velocity of the satellite after the collision.

b) Find the final velocity of the planet.

Answer 1

Hi, Jake. The solution is below.

To solve this problem we will use the law of conservation of linear momentum and the law of conservation of kinetic energy.  Let’s select the final direction of motion of the satellite as positive. Then the law of conservation of linear momentum looks like this

                                         - m_sv_si + m_pv_pi =   m_sv_sf + m_pv_pf                      (1)

For kinetic energy we have

                                              ½ m_sv_si² + ½ m_pv_pi² = ½ m_sv_sf² + ½ m_pv_pf²      (2)

Let’s multiply equation (2) by 2. Then

                                                     m_sv_si² + m_pv_pi² =   m_sv_sf² +  m_pv_pf²            (3)

Rewriting equation so that terms for satellite on the left side of equation and for planet - on the right, we get

                                           - m_sv_{si -} m_sv_sf = - m_pv_pi + m_pv_pf                               (4)

                                            m_sv_si² - m_sv_sf² = m_pv_pf² - m_pv_pi²                          (5)

When we divide equation (5) on equation (4), we obtain

                                              - v_si + v_sf = v_pf  + v_pi                                                                    (6)

After multiplication of the equation (6) by m_s we have

                                           - m_sv_si + m_sv_sf = m_sv_pf + m_sv_pi                                          (7)       

 Adding it to (4) after simplification, we get

                                        - 2 m_sv_si  = (m_s - m_p) v_pi + (m_p + m_s)v_pf

For final velocity of the planet, we obtain

                                                     v_pf = (2 m_sv_si + (m_p – m_s) v_pi )/( m_p + m_s)          (*)

Switching indices, we will have for final velocity of the satellite

                                                     v_sf = (2 m_pv_pi + (m_s – m_p) v_si)/( m_p + m_s)     (**) 

    

The mass of the satellite is negligible compare to the mass of the planet (m_s << m_p), so these velocities reduced to

                                                    v_pf  ≈ v_pi

                                                      v_sf ≈ 2v_pi - v_si

It means that magnitude and direction of the velocity of the planet is approximately the same. But the final velocity of the satellite is changing. It means not only magnitude, but also a direction. what is very important for satellite motion.

Usually the slingshot is used not to change the direction of the satellite to the opposite, but to change it by 90 degrees with respect to the sun. In this case not only the direction is changing but we use gravitational field of the planet to increase the speed of the satellite.