91Ó°ÊÓ

Kinematic equations are essential tools in physics to describe the motion of objects. When working with problems involving motion under constant acceleration, such as the fall of a hailstone due to gravity, these equations become very useful. In this exercise, we focus on finding the final velocity of the hailstone just before it hits the ground.

Here, we're assuming an initial velocity of zero since the hailstone starts from rest and a constant acceleration due to gravity which is approximately 9.8 m/s² near the surface of the Earth. Using the kinematic equation:

• \( v^2 = v_0^2 + 2a d \)

Where:

• \( v_0 \) is the initial velocity
• \( a \) is the acceleration due to gravity
  
• \( d \) is the distance or height
  

We compute the final velocity \( v \). It's a straightforward computation if you methodically identify and insert the given values. This equation helps you predict the motion of an object based solely on its initial state without needing to observe every moment of its journey.

Energy conservation is a core principle in physics, stating that energy in a closed system remains constant. In this exercise, we assume no air resistance, thus energy conservation aids in solving for the final velocity of the hailstone as it transforms potential energy at a height into kinetic energy.

The potential energy (PE) at the starting point is given by:

• \( ext{PE} = mgh \)

Where:

• \( m \) is the mass of the hailstone,
• \( g \), the gravitational acceleration,
• \( h \), the height from which it falls.

Kinetic energy (KE) upon reaching the ground is:

• \( ext{KE} = \frac{1}{2}mv^2 \)

Since energy is conserved:

• \( mgh = \frac{1}{2}mv^2 \)
• Solving for \( v \), the mass \( m \) cancels.

This clear transformation of energy offers a streamlined way to deduce physical results, granting insight into how potential energy translates into kinetic energy during the fall.

Air resistance, often referred to as drag, significantly alters the motion of falling objects like hailstones. Although theory calculations show velocities up to 99 m/s, in the real world, hailstones never fall this fast. This deceleration occurs because air slows the fall.

The force of air resistance builds as the hailstone accelerates, reaching a point where it equals the gravitational pull. This equilibrium halts further acceleration, causing the hailstone to fall at a terminal velocity instead. The terminal velocity is notably slower than the theoretical speed calculated without air resistance.

Consequently, while potential energy should convert into kinetic energy without any loss, the presence of drag shifts some of this energy into other forms, such as heat due to friction. This real-world consideration is vital in accurately understanding how energy and force dynamics influence object speeds in practical scenarios.