Question

a ball is attached to one end of a wire, the other end being fastened to the ceiling. the wire is held horizontal, and the ball is relea from rest (see the drawing). it swings downward and strikes a block initially at rest on a horizontal frictionless surface. air resista negligible, and the collision is elastic. the masses of the ball and block are, respectively, 1.6 kg and 2.4 kg, and the length of the wi 1.40 m. find the velocity (magnitude and direction) of the ball (a) just before the collision, and (b) just after the collision.

Explanation:

Step1: Calculate velocity before collision (using conservation of mechanical energy)

The ball is released from rest, so its initial kinetic energy \(K_{i} = 0\) and initial potential energy \(U_{i}=m_{1}gL\) (where \(m_{1}\) is mass of ball, \(g = 9.8\ m/s^{2}\), \(L\) is length of wire). Just before collision, potential energy \(U_{f}=0\) and kinetic energy \(K_{f}=\frac{1}{2}m_{1}v_{1}^{2}\).
By conservation of mechanical energy \(K_{i}+U_{i}=K_{f}+U_{f}\), so \(m_{1}gL=\frac{1}{2}m_{1}v_{1}^{2}\).
Cancel \(m_{1}\) (since \(m_{1}
eq0\)) and solve for \(v_{1}\):

$$v_{1}=\sqrt{2gL}$$

Substitute \(g = 9.8\ m/s^{2}\) and \(L = 1.40\ m\):

$$v_{1}=\sqrt{2\times9.8\times1.40}=\sqrt{27.44}\approx5.24\ m/s$$

Direction is along \(+x\) - axis (as per coordinate system in the problem).

Step2: Calculate velocity after collision (using elastic - collision formulas)

For an elastic collision (\(e = 1\)), the formula for the velocity of the first object (\(v_{1f}\)) after collision is \(v_{1f}=\frac{m_{1}-m_{2}}{m_{1}+m_{2}}v_{1}\) (where \(m_{1}\) is mass of ball, \(m_{2}\) is mass of block, \(v_{1}\) is velocity of ball before collision).
Given \(m_{1}=1.6\ kg\), \(m_{2}=2.4\ kg\), \(v_{1}=5.24\ m/s\)

$$v_{1f}=\frac{1.6 - 2.4}{1.6+2.4}\times5.24=\frac{- 0.8}{4}\times5.24=- 1.05\ m/s$$

The negative sign indicates the direction is along \(-x\) - axis.

Answer:

(a) The velocity of the ball just before the collision is \(5.24\ m/s\) in the \(+x\) direction.
(b) The velocity of the ball just after the collision is \(-1.05\ m/s\) (or \(1.05\ m/s\) in the \(-x\) direction).