A 26 g rifle bullet traveling 240 m/s buries itself in a 3.5 kg pendulum hanging on a 2.5 m long string, which makes the pendulum swing upward in an arc.

Question

Determine the vertical and horizontal components of the pendulum's displacement.

Express your answers using two significant figures. Enter your answers numerically separated by a comma.

Answer 1

Momentum is conserved here.

The initial momentum before impact is Mi * Vi = 0.026 * 240 = 6.24 Nt - sec

The final momentum after impact is Mf * Vf = 3.526 * Vf = Vi * Mi or Vf = Mi * Vi = 6.24 / 3.526 = 1.77 m/sec

After impact, energy is conserved.

As the pendulum swings the kinetic energy of the combined mass is converted to potential energy. When the mass stops at the top of its swing the potential energy will equal to the kinetic energy at the time of impact.\:

(1/2) * Mf * Vf^2 = Mf * g * h

where h is the (vertical displacement) height reached at the top of the swing and g is the acceleration due to gravity. We can solve this for h:

h = (1/2) * (1.77m/sec)^2 / (9.81 m/sec/sec) = 0.16m

Knowing the height we can find the angle of the pendulum at the top of the swing:

A = arccos[(2.5 - 0.16) / 2.5] or A = 20.61degrees

From A we get the displacement along the horizontal axis:

x = 2.5 * sin(A) = 0.88m

So, the displacement is 0.88m, 0.16m