6 CONSERVATION OF MOMENTUM 6.7 Collisions in 2 - dimensionsm (^1) v
i1
i
m 1 + m 2
vf
f (^) x
vi2
m 2
y
Figure 56: A totally inelastic collision in 2 - dimensions.
For the special case of an elastic collision, we can equate the total kinetic ener-
gies of the two objects before and after the collision. Hence, we obtain
1
m v 2 =
1
m v 2 +
1
m^ v^2.^ (6.63)^
2
(^1) i1
2
(^1) f1
2
(^2) f2
Given the initial conditions (i.e., m 1 , m 2 , and vi1), we have a system of three
equations [i.e., Eqs. (6.61), (6.62), and (6.63)] and four unknowns (i.e., θ 1 , θ 2 ,
vf1, and vf2). Clearly, we cannot uniquely solve such a system without being given
additional information: e.g., the direction of motion or speed of one of the objects
after the collision.
Figure 56 shows a 2-dimensional totally inelastic collision. In this case, the
first object, mass m 1 , initially moves along the x-axis with speed vi1. On the other
hand, the second object, mass m 2 , initially moves at an angle θi to the x-axis with
speed vi2. After the collision, the two objects stick together and move off at an
angle θf to the x-axis with speed vf. Momentum conservation along the x-axis
yields
m 1 vi1 + m 2 vi2 cos θi = (m 1 + m 2 ) vf cos θf. (6.64)
Likewise, momentum conservation along the y-axis gives
m 2 vi2 sin θi = (m 1 + m 2 ) vf sin θf. (6.65)
Given the initial conditions (i.e., m 1 , m 2 , vi1, vi2, and θi), we have a system of