Two balls of equal mass and radius collide where their velocities are parallel b
ID: 2998153 • Letter: T
Question
Two balls of equal mass and radius collide where their velocities are parallel but not in
line. The perpendicular distance between their initial velocities is the same as their
radius. If the collision is perfectly elastic, find the velocity of both balls after the
collision. The initial speed of ball A is 10 m/s and the speed of ball B is 5.0 m/s.
Given: vA1, vB1, d = rA = rB, mA = mB, perfectly elastic collision
Find: vA2, vB2
The line of collision is along the radii of the two balls.
cos ? = d/2r
? = sin-1(d/2r) = sin-1(1/2) = 30degrees
Explanation / Answer
The problem can be solved as follows:
Let us resolve the velocity components of both the velocities in the mentioned x-y coordinate system which would give
vA1 = 5*sqrt(3) i + 5 j m/s
vB1 = -(5/2)*sqrt(3) i - (5/2) j m/s
Further, let's have linear momentum conservation after the elastic collision (Relative velocity of approach of the two bodies in the x-direction must be equal to the relative velocity of separation of the two bodies in the x-direction because of the energy conservation).
The linear momentum conservation in the x-direction would yield along with the elastic collision condition mentioned above:
vA2_x = -(5/2)*sqrt(3) m/s
vB2_x = 5*sqrt(3) m/s
The y-components remain the same because there has been no force or impulse applied in this direction.
Hence, the final velocities become:
vA2 = -(5/2)*sqrt(3) i + 5 j m/s
vB2 = 5*sqrt(3) i + -(5/2) j m/s
Alternatively, one can always use the energy conservation instead of the elastic collision condition mentioned above to get the same answer.