Question

A lab student fires a 0.5 kg projectilewith speed 10 m/s into a ballistic pendulum with a mass of 1.5 kg attached to amassless string 1.0 m in length. What angle does the string makes with the vertical if the pendulum rises to a heightafter impact?​

Answer 1

Given:

A lab student fires a 0.5 kg projectile with speed 10 m/s into a ballistic pendulum with a mass of 1.5 kg attached to a massless string 1.0 m in length.

To find:

Angle made by string with vertical?

Calculation:

Applying CONSERVATION OF LINEAR MOMENTUM:

$\rm \: (0.5 \times 10) + (1.5 \times 0) = (0.5 + 1.5)v$

$\rm \implies\: 5 = 2v$

$\rm \implies\: v = 2.5 \: m {s}^{ - 1}$

Now, the pendulum bob will follow the CONSERVATION OF MECHANICAL ENERGY principle:

$\rm \: PE1 + KE1 = PE2 + KE2$

$\rm \implies \: 0 + \dfrac{1}{2} (2) {(2.5)}^{2} = (2)gh + 0$

$\rm \implies \: \dfrac{1}{2} (2) {(2.5)}^{2} = (2)gh$

$\rm \implies \: 6.25 = 20h$

$\rm \implies \: h = 0.3125 \: m$

Now, calculate the angle:

$\rm \: \cos( \theta) = \dfrac{1 - 0.3125}{1}$

$\rm \implies \: \cos( \theta) = \dfrac{0.6875}{1}$

$\rm \implies \: \cos( \theta) = 0.6875$

$\rm \implies \: \theta = {43.4}^{ \circ}$

So, angle made with vertical is 43.4°.