91Ó°ÊÓ

The kinematic equations are essential tools to solve problems involving motion with constant acceleration. They relate the basic quantities involved in motion, such as displacement, initial velocity, final velocity, acceleration, and time. Here are a few important kinematic equations:

• \( s = ut + \frac{1}{2}at^2 \) - This equation helps us find displacement (\( s \)) when we know initial speed (\( u \)), time (\( t \)), and acceleration (\( a \)).
• \( v = u + at \) - This equation is used to find the final speed (\( v \)), using initial speed, time, and acceleration.
• \( v^2 = u^2 + 2as \) - This equations helps calculate final velocity without considering time.

By using these equations, you can answer various questions about an object's motion, such as in the original exercise of determining the speed and acceleration of a car as it passes through checkpoints.

Initial speed, often denoted as \( u \), refers to the velocity at which an object begins its motion. Understanding the initial speed is important because it serves as a starting point for calculating other components of motion in uniformly accelerated motion.
In the original exercise, the car's initial speed was given as \( 5 \, \text{m/s} \). This value is crucial as it is used in both the kinematic equation for displacement \( s = ut + \frac{1}{2}at^2 \) and the equation for final velocity \( v = u + at \).
Knowing the initial speed allows us to predict how the object's speed and position change over time when acceleration is constant.

Final speed, denoted as \( v \), is a measure of how fast an object is moving at the end of a period of time or distance. In scenarios involving uniform acceleration, final speed can be calculated directly using the kinematic equation:

• \( v = u + at \)

In the provided exercise, after calculating the car's acceleration, we used this equation to find that the final speed at the second checkpoint was \( 10 \, \text{m/s} \).
Final speed is important for both understanding motion and for planning, such as determining whether a car can stop within a given distance. Knowing this speed helps to paint the entire picture of an object's motion under constant acceleration.

Acceleration calculation is a fundamental aspect of understanding uniformly accelerated motion. Acceleration (\( a \)) is the rate of change of velocity over time. To find acceleration in the original exercise, we used the kinematic equation that relates distance, initial speed, time, and acceleration:

• \( s = ut + \frac{1}{2}at^2 \)

By substituting the known values of \( s = 30 \, \text{m} \), \( u = 5 \, \text{m/s} \), and \( t = 4 \, \text{s} \), we solved for \( a \) and found it to be \( 1.25 \, \text{m/s}^2 \).
The calculated acceleration tells us how rapidly the car's speed increases over time, providing insight into the motion characteristics of the object involved. Calculating acceleration is vital for predicting an object's future velocity and position in uniformly accelerated motion.