91Ó°ÊÓ

Projectile motion involves objects moving in parabolic paths under the influence of gravity alone. In this problem, the rocket initially moves vertically until reaching a maximum height of 80 meters. At this point, gravity pulls the rocket back towards the ground.

The explosion that fragments the rocket introduces horizontal velocity to the pieces. Both fragments, upon the explosion, start a projectile motion with no initial vertical velocity, as the explosion occurs at the maximum height. They move horizontally while descending so they essentially follow an arc back to the ground.

In this problem, air resistance is ignored to simplify calculations, making the motion symmetric and allowing us to predict that both fragments land simultaneously, continuing their motion solely under gravity's influence. To comprehend how the fragments travel and land, understanding projectile motion, where acceleration affects solely the vertical motion as gravity, is essential.

Kinetic energy is the energy of motion, which means any object in motion possesses it. The key equation for kinetic energy is \( KE = \frac{1}{2} m v^2 \), where \( m \) represents mass and \( v \) represents velocity.

When the rocket explodes, chemical energy stored becomes kinetic energy, distributed between the two pieces. This shift from potential to kinetic energy is a typical transformation observed in various physical scenarios, particularly after forces act on otherwise stationary objects.

Here, the energy imparted by the explosion totals 860 J. Since kinetic energy is dependent on both mass and velocity, the lighter piece, \( m_1 = 0.28 \, kg \), receives more energy relative to its mass than the heavier piece, \( m_2 = 1.40 \, kg \). However, this does mean it moves faster, elucidating the inverse proportion between mass and speed within a kinetic energy framework. It showcases how lighter masses need less energy to achieve the same kinetic energy as heavier masses. Understandably, the energies translate to measurable speeds using the kinetic energy equation.

Momentum is a crucial concept in physics, representing the product of the mass and velocity of an object, expressed via \( p = mv \). Conservation of momentum states that in an explosion or collision without external force influence, the total momentum remains constant.

In this exercise, before the rocket explodes, it possesses zero net momentum at the peak of its trajectory (as its vertical velocity is zero). During the explosion, even though energy is transformed into kinetic energy of the fragments, the total momentum remains unchanged.

The two fragments must, consequently, maintain a zero net momentum system horizontally. Thus, if one fragment moves in one direction, the other must move in the opposite direction at a velocity such that their momenta cancel out: \( m_1v_1 + m_2v_2 = 0 \). This overall conservation ensures that while each fragment moves independently after separation, their mutual effects on total momentum cancel, explaining their paths and respective velocities post-explosion.