91Ó°ÊÓ

One of the first steps in solving kinematics problems is often determining the initial velocity. The initial velocity is the speed and direction of an object at the beginning of a time interval. In the given exercise, we are asked to find the initial velocity of a sports car moving at a constant velocity for a certain distance.
Since the car travels 120 meters in 5 seconds at constant velocity, we can use the formula:

• \( v = \frac{d}{t} \)

This formula implies dividing the distance \(d\) covered by the time \(t\) it took. Plugging in the given values, we have:

• \( v = \frac{120 \text{ m}}{5.0 \text{ s}} = 24 \text{ m/s} \)

This confirms that the initial velocity \( v_i \) is 24 meters per second. Understanding how to calculate this is crucial, as it sets the foundation for further calculations about the car's motion.

Constant acceleration refers to the change in velocity over time, which remains uniform throughout the period considered. In the context of the problem, the car experiences constant acceleration as it brakes to a stop.
When the car decelerates uniformly over 4 seconds, our task is to use the initial velocity to find this acceleration. The formula typically used is:

• \( a = \frac{v_f - v_i}{t} \)

Here, \( v_f \) is the final velocity and \( t \) is the time. In our scenario, the final velocity \( v_f \) is 0 m/s (since the car stops), the initial velocity \( v_i \) is 24 m/s, and the time \( t \) is 4.0 seconds.
Substituting these values gives:

• \( a = \frac{0 \text{ m/s} - 24 \text{ m/s}}{4.0 \text{ s}} = -6 \text{ m/s}^2 \)

The negative sign indicates the acceleration is in the opposite direction to the velocity, also known as deceleration. However, when asked for its magnitude, we simply use the positive value, which is 6 m/s². This represents how quickly the car is reducing its speed.

The velocity-time relation describes how an object's velocity changes over time under constant acceleration. It is expressed mathematically as:

• \( v_f = v_i + at \)

In our example, understanding this relation helps us determine how the car's velocity decreases while braking. With an initial velocity \(v_i\) of 24 m/s, the car's velocity decreases uniformly by 6 m/s². It stops completely, reaching a final velocity \(v_f\) of 0 m/s at the end of 4 seconds.
This relationship not only indicates the final velocity but also the slope of the velocity-time graph. As long as acceleration is constant, the velocity-time graph is a straight line. The slope of this line represents the acceleration.
Furthermore, this concept is useful when planning journey times and understanding vehicle motions. It provides insight into the effect of acceleration and deceleration phases in everyday travel scenarios.