91Ó°ÊÓ

Kinematic equations are essential tools in physics to describe the motion of objects. They relate the motion variables—such as displacement, velocity, acceleration, and time—allowing us to calculate one when the others are known. In this scenario, we use kinematic equations to determine the motion of both the airplane and the bullet shot at it.
The standard form of kinematic equations includes:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

In the problem, the airplane's position over time is given by the equation: \[ x_{plane}(t) = 1000 + 367t + \frac{1}{2}a t^2 \] Here, \( 1000 \) m is the initial position, \( 367 \) m/s is the initial velocity, and \( a \) is the unknown acceleration. Meanwhile, the bullet's motion is described using a simpler equation since it moves upward from the ground without additional acceleration. Thus, \[ x_{bullet}(t) = 567t \] Understanding how to apply these equations in different scenarios allows us to predict the future positions of moving objects.

A quadratic equation is a critical part of determining the possible meeting point of the bullet and the airplane. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] Solving a quadratic equation helps us find the values of \( x \) that satisfy the equation. In our context, \( x \) represents the time \( t \) at which the positions of the airplane and bullet are equal.
We derive a quadratic equation by setting the position equations of the aircraft and bullet equal: \[ 1000 + 367t + \frac{1}{2} at^2 = 567t \] This simplifies into: \[ \frac{1}{2}at^2 + (367 - 567)t + 1000 = 0 \] Now, solving this yields the conditions under which the airplane could avoid being hit. The crucial point here is the discriminant of the quadratic equation:\[ b^2 - 4ac \] If it is negative, the quadratic equation has no real solutions for \( t \), meaning the airplane's position never coincides with the bullet's, ensuring safety.

Relative velocity is the speed of one object as observed from another moving object. It plays a key role in analyzing scenarios where multiple objects are moving, such as our case with the airplane and the bullet.
For the airplane to escape the bullet, its effective speed upward must be greater than the bullet's speed or such that they never meet. The concept of relative velocity can be framed as:

• When two objects move towards each other, their relative speed is the sum of their speeds.
• When two objects move in the same direction, their relative speed is the difference between their speeds.

In the given problem, the relative motion concept helps us understand how quickly the bullet closes in on the plane. With an upward speed of the airplane and the bullet moving in the same direction initially, it's crucial to determine if the airplane's acceleration can increase its relative velocity enough to escape being hit. To find this, we analyze the quadratic equation derived from their motion, ultimately finding the minimum acceleration needed to maintain a safe distance.