91Ó°ÊÓ

Free fall occurs when an object is moving only under the influence of gravity. When the hawk releases the mouse, the mouse starts experiencing free fall. There is no air resistance assumed in this scenario, so the only force acting on it is gravity.
During free fall, an object accelerates at the rate of gravitational acceleration, commonly approximated as \( g = 9.8 \, \text{m/s}^2 \).
In this exercise, the mouse falls from a height of 200 m and almost reaches the ground, covering 197 meters.
As the initial velocity of the mouse is 0 m/s (as it is only dropped and not pushed), we can compute the time of free fall using the equation of motion:

• 
  \( s = ut + \frac{1}{2}gt^2 \)
  

Solving for time \( t \), the equation simplifies to \( t = \sqrt\frac{2s}{g} \).
This calculation gives us the time span during which the mouse is enjoying its brief flight.

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. It provides languages, equations, and principles to describe how things move.
For this problem, we apply kinematic equations to both the mouse and the hawk. The mouse starts with a vertical initial velocity of 0 m/s, and the hawk has a horizontal constant velocity of 10.0 m/s.
As the hawk flies horizontally for 2 seconds after releasing the mouse, its horizontal motion can be described using:

• 
  \( d = ut \)
  

where \( u = 10 \, \text{m/s} \) and \( t = 2 \, \text{s} \). This gives us the horizontal distance the hawk has already covered. Combining this with the vertical motion of the mouse helps in understanding their spatial dynamics.

Trigonometry deals with the properties and relationships of triangles. It is particularly useful in projectile motion problems due to the frequent presence of right-angled triangles.
In this exercise, when the hawk dives to capture the mouse, it forms a right-angled triangle with the horizontal distance covered forming the base and the height difference forming the vertical part.
Using the Pythagorean theorem, we can determine the total distance the hawk covers during its descent:

• 
  \( D = \sqrt{(d + 10.0t)^2 + s^2} \)
  

To find the angle of descent, trigonometry provides the tangent function:

• \( \tan(\theta) = \frac{s}{(d + 10.0t)} \)
  
• \( \theta = \arctan\left(\frac{s}{d + 10.0t}\right) \)
  

These calculations aid in determining the angle at which the hawk descends to catch its prey.

Newton's Laws of Motion are fundamental to understanding how forces and motion interact. They provide the foundation for analyzing the motion of objects.
In the context of this problem, the concept primarily applies to the hawk, which continues its horizontal path at a steady speed - demonstrating Newton's First Law: an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Moreover, when the mouse falls under gravity, Newton's Laws elucidate the unbalanced force acting on it, causing it to accelerate downwards as per the Second Law, \( F = ma \). Here, gravity is the force and mass is that of the mouse, resulting in an acceleration equal to g.

Newton's principles assist in setting up and solving the equations of motion, allowing us to understand the motion's kinematic and dynamic aspects thoroughly.