Question

A 29-g rifle bullet traveling 220 m/s embeds itself in a 3.0-kg pendulum hanging on a 2.7-m-long string, which makes the pendulum swing upward in an arc. (Figure 1)

A) Determine the vertical component of the pendulum's maximum displacement.

B) Determine the horizontal component of the pendulum's maximum displacement.

Answer 1

A] By momentum conservation, Let the velocity after it embads be v

m1u1 + m2u2 = [m1+m2]v

0.029*220 + 3*0 = 3.029v

v = 0.029*220/3.029 = 2.10 m/s

Now by energy conservation, Let maximum angle is theta,

0.5* [m1+m2]v^2 =  [m1+m2]*g*L[1 - cos theta]

0.5* 2.10^2 = 9.8*2.7*[1 - cos theta] removing mass from both side

0.5* 2.10^2/(9.8*2.7) = [1 - cos theta]

  0.08333 = 1- cos theta

cos theta = 1 - 0.08333 = 0.91667

theta = arccos 0.91667 = 23.56 degree

Vertical displacement = L*(1 - cos theta) = 2.7*0.08333 = 0.225 m answer

B] horizontal displacement = 2.7* sin 23.56 degree = 1.079 answer