Question

An explosion blows a rock into three parts. two pieces go off at right angles to each other; 1.0 kg piece with velocity of 12 m/s and other 2.0 kg piece with a velocity of 8 m/s. if third piece flies off with the velocity of 40 m/s, compute the mass of third piece.

Answer 1

Hey there!

Let the masses of three pieces be m₁ , m₂ , m₃ respectively.

And, Let their velocities be v₁ , v₂ , v₃ respectively.

Given,

m₁ = 1.0  kg

m₂ = 2.0 kg

v₁ = 12 m/s

v₂ = 8 m/s

v₃ = 40 m/s

Let v₁ and v₂ be directed along x-axes and y- axes respectively;

And v₃ be directed as shown in the figure attached.

By principle of conservation of momentum, initial momentum is zero.

So, along x - axis, m₁v₁  - m₃v₃ cos Ф = 0

and, along y - axis, m₂v₂ - m₃v₃ sin Ф = 0

Hence,

m₁v₁ = m₃v₃ cos Ф ......... (i)

m₂v₂ = m₃v₃ sin Ф ......... (ii)

Squaring and Adding both the equations we get:

$\Large {m_1^2 v_1^2 + m_2^2v_2^2 = m_3^2v_3^2}$

Then,

$m_3^2 = \dfrac{1^2.(12)^2 + (2^2).8^2}{(40)^2}\\\\\boxed{\sf m_3 = 0.5 \ kg}$

Hence, mass of third piece = 0.5 kg.