The 400-kg satellite S traveling at 7 km/s is hit by a 1-kg meteor M. The meteor is embedded in the satellite by the impact. What would the magnitude of the velocity of the meteor need to be to cause the angle f3 between the original path of the satellite and the path of the center of mass of the combined satellite and meteor after the impact to be 0.5°? What is the magnitude of the velocity of the center of mass after the impact?

a) v_m = 34.26 km / s      

b) v_ms = 6.9224 km/s

Explanation:

Given:

- The mass of the satellite m_s =400 kg

- The velocity of satellite v_s = 7 km /s

- The mass of the meteor m_m = 1 kg

- Angle between meteor and satellite impact a = 135 degrees.

- Angle between original path and combined mass after impact B = 0.5 degrees

Find:

- What would the magnitude of the velocity of the meteor need to be to cause the angle a.

- What is the magnitude of the velocity of the center of mass after the impact?

Solution:

- We will apply the principle of conservation of momentum before and after the impact to be equal or total momentum to be zero. We will divide our momentum equation in two components.

- The conservation of momentum along the satellite path:

                                              P_i = P_f

           m_s*v_s + m_m*v_m*cos(135) = (m_s + m_m)*v_ms*cos(0.5)   ... Eq 1

Where,

- v_m,s is the combined velocity of satellite and meteor.

- Similarly, the conservation of momentum normal to the satellite path:

                                               P_i = P_f

                    m_m*v_m*sin(135) = (m_s + m_m)*v_ms*sin(0.5)   Eq 2

Where,

- v_m,s is the combined velocity of satellite and meteor.

- Use the derived results in both components, Divide Eq 2 by Eq 1:

        tan(0.5)*[m_s*v_s + m_m*v_m*cos(135)] = m_m*v_m*sin(135)

        tan(0.5)*m_s*v_s =  m_m*v_m*( sin(135) - cos(135)*tan(0.5))

       v_m =  [tan(0.5)*m_s*v_s] / [m_m*( sin(135) - cos(135)*tan(0.5))]

Plug in the values:

           v_m =  [tan(0.5)*400*7] / [1*( sin(135) - cos(135)*tan(0.5))]

                                      v_m = 34.26 km / s      

- Now, use one of the equations and find v_ms, using Eq 2:

                 m_m*v_m*sin(135) = (m_s + m_m)*v_ms*sin(0.5)

                 v_ms = [m_m*v_m*sin(135)] / [sin(0.5)*(m_s + m_m)]

Plug in the values:

                 v_ms = [1*34.26*sin(135)] / [sin(0.5)*(400 + 1)]  

                                      v_ms = 6.9224 km/s

the total distance traveled by the car is

30km + 49km = 79 km .