Respuesta :
Answer:
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
