91Ó°ÊÓ

Kinematic equations are formulas used to describe the motion of objects, like a bolt falling from a bridge, under constant acceleration. These equations help us calculate various aspects of motion, such as distance, time, velocity, and acceleration, without requiring detailed knowledge of the forces involved. They are particularly useful in situations like free fall, where the only force is gravity. There are four commonly used kinematic equations:

• 1. Final velocity: \( v = u + at \)
• 2. Displacement: \( s = ut + \frac{1}{2}at^2 \)
• 3. Final velocity squared: \( v^2 = u^2 + 2as \)
• 4. Displacement without final velocity: \( s = \frac{u + v}{2} \times t \)

Here,

• \( v \) is the final velocity.
• \( u \) is the initial velocity.
• \( a \) is the acceleration.
• \( t \) is the time.
• \( s \) is the displacement.

These relations allow us to solve problems about objects moving in a straight line with constant acceleration, like our bolt falling under the influence of gravity.

The acceleration due to gravity is a key concept when analyzing free fall motion. On Earth, this acceleration is approximately \( 9.8 \ m/s^2 \), meaning that any object in free fall will increase its velocity by \( 9.8 \ m/s \) every second, assuming no air resistance.

This value is constant irrespective of the object's mass. So when the bolt falls from the bridge, it continuously accelerates towards the ground at this rate. This acceleration affects how quickly the object reaches certain speeds and how much time it takes to travel certain distances.

In our exercise, we use this value to determine both the velocities at different stages of the bolt's fall and the time required for it to pass through specific portions of the fall.

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. They often arise in physics when dealing with equations involving time and acceleration, due to the quadratic relationship between these quantities.

In the context of the free-fall problem, the equation \( s = ut + \frac{1}{2}at^2 \) becomes a quadratic equation when solving for time \( t \). After substituting known values and rearranging, the equation looks like this:

• \( 4.9t^2 + 37.56t - 18 = 0 \)

We solve these equations using the quadratic formula:

• \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

For our bolt problem, solving this equation gives us the time it takes to fall the last 18 meters. Understanding how to solve quadratic equations is critical in physics for finding these critical time intervals.