Understanding Acceleration through Kinematic Equations: A Practical Example

Accelerating from one speed to another over a specific distance is a common scenario in everyday life, particularly for vehicles. In this article, we will delve into a practical example to understand how to calculate the average acceleration of a car that increases its speed from 40 m/s to 80 m/s over a distance of 200 meters. We will use the kinematic equations to solve for the acceleration and explore the significance of this value in the context of motion and physics.

1. Identifying Known and Unknown Quantities

Initial velocity (u): 40 m/s Final velocity (v): 80 m/s Acceleration (a): To be determined Distance (s): 200 meters

2. Applying the Kinematic Equation

Step 1: We will use the following kinematic equation, which relates the final velocity, initial velocity, acceleration, and distance covered:

v2 u2 2as

Step 2: Rearrange the equation to solve for acceleration (a):

a (v2 - u2) / 2s

3. Substituting Values and Calculating Acceleration

Given values:

Initial velocity (u) 40 m/s Final velocity (v) 80 m/s Distance (s) 200 meters

Step 3: Substitute the given values into the equation:

a  (802 - 402) / (2 × 200)

Step 4: Simplify the expression:

a (6400 - 1600) / 400

a 4800 / 400

a 12 m/s2

Therefore, the average acceleration of the car is 12 meters per second squared.

4. The Significance of the Acceleration Value

The result is interpreted as follows: if the car was initially moving at 40 m/s and it was able to accelerate to 80 m/s over a distance of 200 meters, the average rate of that acceleration was 12 m/s2. This value is crucial in understanding the car's performance, safety, and power output.

5. Alternative Methods: Work-Energy Theorem

Another approach to finding the acceleration is through the work-energy theorem. The work done by a force (F ma) over a distance (x) is equal to the change in kinetic energy:

W Fx 1/2mv2 - 1/2mv2

The kinetic energy change can be represented as:

1/2mv2 - 1/2mv2 1/2m (802 - 402)

1/2m (6400 - 1600)

1/2m × 4800

Since F ma, and the work done is equal to the force times the distance:

1/2m × 4800 ma × 200

1/2 × 4800 a × 200

a 4800 / 400 12 m/s2

This method provides a different perspective but reaches the same conclusion, namely that the car's average acceleration is 12 m/s2.

Conclusion

In conclusion, calculating the acceleration of a car in motion, from 40 m/s to 80 m/s over 200 meters, yields an average acceleration of 12 m/s2. This value is significant for understanding the car's performance and can be calculated using both kinematic equations and the work-energy theorem. Understanding these principles helps in various applications, from automotive design to sports performance.