What is the Velocity of the Ball After Impact?

[GayYoda]

Homework Statement

Instead of using a ballistic pendulum, a bullet with velocity u is fired at a stationary solid ball resting on a surface. If the bullet deflects at an angle of 30◦ to its original path and the ball is nine times more massive than the bullet, what is the velocity of the ball after the impact? You should assume that the ball only moves horizontally and does not bounce or lift from the surface.

Homework Equations

p=mv
KE=mv^2/2

The Attempt at a Solution

Conservation of momentum: mu=9mv_2-mv_1cos(30) => u=9v_2-v_1cos(30)
Conservation of energy: mu^2/2=mv_1^2+9mv_2^/2 => u^2=v_1^2+9v_2^2
I'm not sure how to combine these equations to get the velocity of the ball (v_2)

 

[haruspex]

mu=9mv_2-mv_1cos(30)

the velocity of the ball (v_2)

You seem to be assuming the direction of the departing ball is the same as the original direction of the bullet.
Momentum is a vector, so there are two directions to consider.

Conservation of energy

What grounds do you have for supposing work is conserved?

 

[kuruman]

I'm not sure how to combine these equations to get the velocity of the ball (v_2)

Do you know how to solve a system of 2 equations and 2 unknowns?
Also, your solution assumes that the surface is frictionless because you do not take into account that the ball might be rolling after the collision. There is no language in the problem stating that this is the case.

 

[haruspex]

the ball might be rolling after the collision.

If the impact takes infinitesimal time then immediately after impact it will only be rotating infinitesimally. It will take time to transition to rolling.
(This is assuming the bullet strikes the ball in the horizontal plane of its mass centre.)

 

[kuruman]

Momentum is a vector, so there are two directions to consider.

Is momentum conserved in the direction perpendicular to the surface?

If the impact takes infinitesimal time then immediately after impact it will only be rotating infinitesimally. It will take time to transition to rolling.

So the problem is asking for the velocity of the ball as if the surface were frictionless. OK.

This is assuming the bullet strikes the ball in the horizontal plane of its mass centre.)

I guess an additional assumption would have to be that the bullet is not fired in a direction parallel to the surface.

 

[haruspex]

Is momentum conserved in the direction perpendicular to the surface?

Your comments have made me realize the question is ambiguous and we have made different interpretations. I took it that the deflection is in the horizontal plane. I think if it meant deflected upwards it would have said so.

So the problem is asking for the velocity of the ball as if the surface were frictionless. OK.

If the deflection is horizontal then friction from the surface is irrelevant unless the coefficient is huge. If the deflection is upwards then there is a vertical component to the impulse and friction becomes important.

 

[kuruman]

I think if it meant deflected upwards it would have said so.

What made me think of upward deflection is the statement, "You should assume that the ball only moves horizontally and does not bounce or lift from the surface." This statement is redundant if all motion takes place in a plane parallel to the surface.

 

[haruspex]

What made me think of upward deflection is the statement, "You should assume that the ball only moves horizontally and does not bounce or lift from the surface." This statement is redundant if all motion takes place in a plane parallel to the surface.

True, but I still find it very odd just to write "deflected" if it meant deflected upwards. Maybe it is not a verbatim copy of the original problem.

Either way, there is insufficient information. It would certainly not be appropriate to assume conservation of work, but the coefficient of restitution need not be zero either.