Introduction

This article explores the principles of linear momentum in the context of two stones of different masses being dropped from different heights. By examining the relationship between their velocities and momenta, we can determine the ratio of their linear momenta just before reaching the ground, neglecting air resistance. This analysis is essential for understanding fundamental concepts in physics, particularly in the areas of mechanics and dynamics.

The Physics Behind the Problem

When two stones of different masses are dropped from heights in the ratio of 4:9, their velocities just before hitting the ground can be evaluated using the principles of kinematics and conservation of energy. The mass and height are crucial factors in determining the velocities and, consequently, the linear momenta of the stones.

Velocity of Falling Stones

The velocity of a mass (m_i) that falls freely under gravity from an initial height (h_i) can be given by the equation:

[ v_i sqrt{2gh_i} ]

where (g) is the acceleration due to gravity. This equation is derived from the kinematic equation:

[ v_i^2 u_i^2 2gh_i ]

Here, (u_i 0) since the stones are dropped from rest.

Deriving Linear Momentum

Linear momentum is the product of mass and velocity. Therefore, the linear momentum (p_i) of the stone with mass (m_i) can be expressed as:

[ p_i m_i cdot v_i m_i cdot sqrt{2gh_i} ]

Substituting (v_i sqrt{2gh_i}) into the equation for linear momentum, we get:

[ p_i sqrt{m_i^2 cdot 2gh_i} ]

Ratio of Linear Momenta

To find the ratio of the linear momenta of the two stones, we need to compare (p_1) and (p_2). The linear momentum for the first stone is:

[ p_1 sqrt{m_1^2 cdot 2gh_1} ]

The linear momentum for the second stone is:

[ p_2 sqrt{m_2^2 cdot 2gh_2} ]

Therefore, the ratio of their linear momenta is given by:

[ frac{p_1}{p_2} sqrt{frac{m_1^2 cdot h_1}{m_2^2 cdot h_2}} ]

Applying Given Ratios

Given that:

[ frac{m_1}{m_2} frac{3}{2} ] and [ frac{h_1}{h_2} frac{4}{9} ]

Substituting these values into the equation for the ratio of momenta:

[ frac{p_1}{p_2} sqrt{frac{frac{3}{2}^2 cdot 4}{frac{4}{9}}} sqrt{frac{9/4 cdot 4}{4/9}} sqrt{1} 1 ]

This result indicates that the linear momenta of the two stones are equal just before they hit the ground.

Conclusion

In this analysis, we have demonstrated that the linear momenta of two falling stones, regardless of their masses and the heights from which they are dropped, are equal just before they reach the ground. This conclusion is derived from the principles of kinematics and the conservation of energy. Understanding these concepts is crucial for solving complex problems in physics and engineering.