Conservation of momentum going insane here

[mdergance3]

From what I remember about momentum conservation is conserved in both x and y directions. Energy must be conserved as well (elastic collision)

If I have a cue ball moving to the right with velocity of 4 m/s and it hits another stationary cue ball with equal mass. The first cue ball is deflected at a velocity of 1.5 m/s @ 45° in the north east direction. What is the velocity and direction of the 2nd cue ball.

I apply conservation of linear momentum, I'm going to negate the masses since they are equal.

4 = 1.5*cos45 + v_x
where v_x is the x component of the 2nd cue ball

v_x = 2.94 m/s

Now the y direction:

1.5*sin45 = -v_y
v_y = 1.06m/s

magnitude of 2nd cue ball's veloctiy = sqrt(1.06^2 + 2.94^2) = 3.125

Now conservation of energy:

.5*m*4^2 = .5*m*3.125^2 + .5*m*1.5^2
16 = 12.01 <-- this is not true, why? what am I doing wrong?

 

[Yuqing]

This happens because the final conditions were specified. Normally, conservation of energy and momentum are used together to determine the final configuration of a collision. In this case, the final conditions were specified and so energy conservation is in a sense independent. You can also view this as the case in a completely inelastic collision, because the course of the collision is already predetermined (they hit each other and then stick together), there is no freedom to explore the conservation of energy here, it must happen that the maximum amount of energy is lost in a completely inelastic collision. In this case, the collision will not be elastic in most cases, especially when the question is prepared carelessly. I remember a question in my first year physics exam where the final energy exceeded the initial (which was quite humorous).

 

[AlephZero]

Does the question say it is a perfectly elastic collision, or are you just assuming that it is?

Your momentum calcs look OK. The system loses some kinetic energy in the collision, so it was not perfectly elastic. There isn't any "real world" problem with that situation.

 

[nasu]

How do you know the values for the final state (velocity and angle)? Are they measured or is just a hypothetical case?

Anyway, for off-center collision, the balls may start to spin around their axes and there will be some rotational kinetic energy.

 

[BruceW]

The crux of the problem is that In your set-up, if the first ball is deflected at 45° at 1.5m/s, then the second ball will be deflected at -45° with the same speed (to conserve momentum), but then the total KE after collision is greater than total KE before.
Therefore, this is impossible for an elastic collision, it can only happen for an inelastic collision.
Of course, on a real pool table, the spin of the balls will have a big effect.

 

[sophiecentaur]

The crux of the problem is that In your set-up, if the first ball is deflected at 45° at 1.5m/s, then the second ball will be deflected at -45° with the same speed (to conserve momentum), but then the total KE after collision is greater than total KE before.
Therefore, this is impossible for an elastic collision, it can only happen for an inelastic collision.
Of course, on a real pool table, the spin of the balls will have a big effect.

Not necessarily 45 degrees - the transverse momentum values would, of course, be equal and opposite but the actual angle of the resultant velocities would also depend upon the longitudinal velocities / momentums.

 

[BruceW]

woops, yeah I think I got that wrong. Its not necessarily both deflected at 45 degrees.
Thanks, sophiecentaur

 

[willem2]

Not necessarily 45 degrees - the transverse momentum values would, of course, be equal and opposite but the actual angle of the resultant velocities would also depend upon the longitudinal velocities / momentums.

If you hit a stationary ball, there's always a 90 degree angle between the directions of the two balls after the collision, so if one angle is +45, the other has to be -45, and the
longitudinal velocities would be the same.

 

[Dadface]

mdergance ,it has been pointed out that this is an inelastic collision and with such collisions total energy is conserved but there is a loss of kinetic energy(KE changed to heat etc)
With elastic collisions KE is conserved but such collisions can only happen with microscopic objects such as atoms and then only under certain conditions.Macroscopic objects like billiard balls can never collide elastically.