Physics Problem A very large mass (for instance, the planet Jupiter) and a very
ID: 1600878 • Letter: P
Question
Physics Problem
A very large mass (for instance, the planet Jupiter) and a very small mass (for instance, the Voyager spacecraft) are moving toward each other at equal speeds (but in opposite directions). After a perfectly elastic gravitational collision in which the spacecraft swings all the way around the planet and departs in the opposite direction from whence it came, find its new speed (as a multiple of its original speed).
[Hint: the masses of the two objects are not given, but at some point in your math you will have to invoke the approximation that (M of Jupiter) >> (m of spacecraft).]
Answer: _______ x original speed
Explanation / Answer
ELASTIC COLLISION
m1 = M kg m2 = m kg
speeds before collision
v1i = v m/s v2i = -v m/s
speeds after collision
v1f = ? v2f = ?
initial momentum before collision
Pi = m1*v1i + m2*v2i
after collision final momentum
Pf = m1*v1f + m2*v2f
from momentum conservation
total momentum is conserved
Pf = Pi
m1*v1i + m2*v2i = m1*v1f + m2*v2f .....(1)
from energy conservation
total kinetic energy before collision = total kinetic energy after collision
KEi = 0.5*m1*v1i^2 + 0.5*m2*v2i^2
KEf = 0.5*m1*v1f^2 + 0.5*m2*v2f^2
KEi = KEf
0.5*m1*v1i^2 + 0.5*m2*v2i^2 = 0.5*m1*v1f^2 + 0.5*m2*v2f^2 .....(2)
solving 1&2
we get
v1f = ((m1-m2)*v1i + (2*m2*v2i))/(m1+m2)
v1f = ( (M-m)*v - (2*m*v))/(M+m)
since M >>>>m
m is neglected
M + m = M-m = M
V1f = M*v/M = v
v2f = ((m2-m1)*v2i + (2*m1*v1i))/(m1+m2)
v2f = ( -(m-M)*v + 2*M*v)/(M+m
v2f = (Mv + 2MV)/M = 3*v
new speed = 3x original speed <<<<----answer