HINTS FOR PROBLEM SET #11

Problem Set #11 – Problem 1

For impact problems,

you need to find where momentum is conserved.

In this problem momentum is conserved:

          1.  for the system of ball A and ball B in all directions.

               (Because there are no external forces acting during the impact.)

          2.  in the tangential direction for ball A.

               (Because there is no friction during the impact.)

          3.  in the tangential direction for ball B.

               (Because there is no friction during the impact.)

Number 1 allows you to apply the Conservation of Momentum

for the system of ball A and ball B in the normal direction:

           m_Av_An  +  m_Bv_Bn  =   m_Av_An' +  m_Bv_Bn'

This is your 1st equation.

Note:   Primed quantities refer to velocities after the impact.

              Unprimed quantities refer to velocities before the impact.

                   Velocities before the impact are known.

Number 2 allows you to apply the Conservation of Momentum

for ball A in the tangential direction:

           m_Av_At   =   m_Av_At'

           v_At'  =  v_At  = v_A sin 40^o = 3.857 m/s

This your 2^nd equation.

Number 3 allows you to apply the Conservation of Momentum

for ball B in the tangential direction.

           m_Av_Bt   =   m_Av_Bt'

           v_Bt'  =  v_Bt  = 0 m/s

This is your 3^rd equation.

Now apply the relative velocity equation

for the impact between the two balls. 

(Recall, this equation is only valid for the normal direction.)

          (v_Bn' - v_An') = e (v_An - v_Bn)

This is your 4^th equation.

These 4 equations allow you to solve for the 4 unknowns:

          1. the tangential component of the velocity of A after the impact, v_At'

          2. the normal component of the velocity of A after the impact, v_An' 

          3. the tangential component of the velocity of B after the impact, v_Bt'

          4. the normal component of the velocity of B after the impact, v_Bn'

Problem Set #11 – Problem 2

For impact problems, you need to find where momentum is conserved.

In this problem momentum is conserved:

          1.  for ball A in the tangential direction.

               (Because there is no friction during the impact.)

          2.  for the system of the ball A and wedge B in the horizontal direction.

               (Because there is no friction at the ground during the impact.)

Number 1 allows you to apply the Conservation of Momentum

for ball A in the tangential direction:

           m_Av_At   =   m_Av_At'

           v_At' = v_At = 0

Note:  Momentum is not conserved

in the tangential direction for the wedge

because the normal force acting on the wedge from the ground

has a component in the tangential direction. 

Number 2 allows you to apply the Conservation of Momentum

to the system in the x (horizontal) direction.

            m_Av_Ax  +  m_Bv_Bx  =   m_Av_Ax' +  m_Bv_Bx'

Call this equation 1.

Now apply the relative velocity equation

for the impact between the ball A and the wedge B. 

(Recall, this  equation is only valid for the normal direction.)

            (v_Bn' - v_An') = e (v_An - v_Bn)

Call this equation 2.

The above equations gives 2 equations in 4 unknowns. 

Since the velocities are know before the impact, the 4 unknowns are:

          1) the velocity of the ball in the x direction after the impact, v_Ax'

          2) the velocity of the wedge in the x direction after the impact, v_Bx'

          3) the velocity of the ball in the n direction after the impact, v_An'

          4)  the velocity of the wedge in the n direction after the impact, v_Bn'

We need two more equations. 

We can get them from trigonometry.

We know that the velocity of the wedge is in the x direction after the impact.

So its component in the n direction is:

              v_Bn' = v_Bx' sinq

This is your 3^rd equation.

We know that velocity of the ball is in the n direction after the impact.

So its component in the x direction is:

              v_Ax' = v_An' sinq

This is your 4^th equation.

Now you can solve for the 4 unknowns.