Question

Problem 2 A 1.0 kg cannonball is fired horizontally with a speed of 200 rn/s at two blocks (each with mass 10.0 kg at rest on a frictionless surface). The ball embeds itself in the first block, and then the ball and block 1 traveling together collide elastically with block 2. All external forces are negligible compared with the net explosion or collision forces. Use the reference frame provided. Neglect the material removed from the block during collision with the ball. Figure 4: Problem 2. (a) For the instant shown in the above three objects (cannonball, block 1, diagram. find the center of mass position for the block 2). (5pt) (b) Find the speed of the ball and block 1 immediately after the first collision. (5pt) (c) What are the kinetic energies of the ball before the first collision and ball + block after the first collision? How much kinetic energy is lost during the first collision? (5pt) (d) Find the speeds of the blocks after the second collision. (5pt) Bonus: If the ball was fired from a canon of mass 1,000 kg, what is the recoil speed of the canon? (3pt)

Answer 1

a) The center mass position is:

$x_{cm} = \frac{m_0x_0+m_1x_1+m_2x_2}{m_0 + m_1 + m_2}$

$x_{cm} = \frac{(1.0)(0)+(10.0)(1)+10(2.0)}{1.0+10.0+10.0}= 1.4286 m$

b) We use the conservation of linear mometum:

$V_1 = \frac{m_0v_0}{m_0 +m_1} = \frac{(1.0)(200)}{1.0+10.0}= 18.18 m/s$

c) we calculate the initial kinetic energy:

final kinetic energy:

d) we have to apply the conservation of linear mometum again:

using given data we get:

also we know that this is a elastic collision:

$e = -\frac{v_2-v_1}{V_2-V_1} = 1$

so:

solving the system we have:

e) we have:

solving for Vcannon:

$V_{cannon} = \frac{m_0v_0}{M_{cannon}} = \frac{1.0(200)}{1000}=0.200 m/s$