Question

two bodies of mass in the ratio 1:3 fall from heights in the ratio 3:4 Find the ratio of their velocities and linear momentum on reaching the ground ​

Answer 1

Final Answer :

Ratio of velocities = √(3/4)

Ratio of linear momenta = 1/√12

Given :

Two bodies of mass in the ratio 1:3 fall from heights in the ratio 3:4

To Find :

the ratio of their velocities and linear momentum on reaching the ground ​

Solution :

Mass , m₁ = 1 kg & Mass , m₂ = 3 kg

Height , s₁ = 3 kg & Height , s₂ = 4 kg

Now use 3rd equation of motion to find final velocity .[ Here Initial velocity is zero ]

⇒ v₁² - u₁² = 2gs₁

⇒ v₁² = 2 (10)(3)

⇒ v₁² = 60

⇒ v₁ = √60

Like wise , v₂ = √80

So ,

Ratio of their velocities ,

$\bf \dfrac{v_1}{v_2}=\dfrac{\sqrt{60}}{\sqrt{80}}\\\\\implies \bf \underline{\bf \dfrac{v_1}{v_2}=\sqrt{\dfrac{3}{4}}}$

Ratio of their linear momenta ,

$\bf \dfrac{p_1}{p_2}=\dfrac{m_1.v_1}{m_2.v_2}\\\\\implies \bf \dfrac{p_1}{p_2}=\dfrac{m_1}{m_2}.\dfrac{v_1}{v_2}\\\\\implies \bf \dfrac{p_1}{p_2}=\dfrac{1}{3}.\sqrt{\dfrac{3}{4}}\\\\\implies \bf \underline{\bf \dfrac{p_1}{p_2}=\dfrac{1}{\sqrt{12}}}$