Often in designing orbits for satellites, people use what is termed a \"gravitat
ID: 1499556 • Letter: O
Question
Often in designing orbits for satellites, people use what is termed a "gravitational slingshot effect." The idea is as follows: 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. See the diagram below: Although the satellite never touches the planet, it is still a considered to be a collision because the planet and satellite interact through gravity. Because gravity is a conservative force, the collision is elastic. Use an x-axis with positive pointing to the right. Solve for the unknowns below algebraically first, then use the following values for the parameters. m_p = 1.70E + 24 kg m_s = 500 kg v_s, I, x = 1.155E+4 m/s v_p, I, x = -7.70E+3 m/s Solve for thermal velocity of the satellite after the collision.Explanation / Answer
mass of planet = m1 = 1.7*10^24 kg mass planet of satellite m2 = 500 kg
speeds before collision
initial speed of planet u1 = -7.7*10^3 m/s initial speed of satellite u2 = 1.155*10^4 m/s
speeds after collision
v1 = ? v2 =
initial momentum before collision
Pi = m1*u1 + m2*u2
after collision final momentum
Pf = m1*v1 + m2*v2
from momentum conservation
total momentum is conserved
Pf = Pi
m1*u1 + m2*u2 = m1*v1 + m2*v2 .....(1)
from energy conservation
total kinetic energy before collision = total kinetic energy after collision
KEi = 0.5*m1*u1^2 + 0.5*m2*u2^2
KEf = 0.5*m1*v1^2 + 0.5*m2*v2^2
KEi = KEf
0.5*m1*u1^2 + 0.5*m2*u2^2 = 0.5*m1*v1^2 + 0.5*m2*v2^2 .....(2)
solving 1&2
we get
v1 = ((m1-m2)*u1 + (2*m2*u2))/(m1+m2)
final speed of planet Vpfx v1 = (-( ((1.7*10^24)-500)*7.7*10^3) + (2*500*1.155*10^4)) /((1.7*10^24)+500) = -7700 m/s
v2 = ((m2-m1)*u2 + (2*m1*u1))/(m1+m2)
v2 = ( ((500-(1.7*10^24))*1.155*10^4) - (2*1.7*10^24*7.7*10^3) ) /((1.7*10^24)+500)
final speed of satellite Vsfx v2 = -26950 m/s
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