Understanding Projectile Motion: Calculating Time and Velocity Using Kinematic Equations

When dealing with objects in motion, particularly those undergoing acceleration, understanding the proper application of kinematic equations is crucial. This article focuses on calculating the time and velocity of an object that starts from rest, accelerates at a constant rate, and covers a specific distance. The question at hand is: If an object initially at rest accelerates at a rate of 4 m/s2, how long will it take the object to travel 200 meters?

Applying Kinematic Equations

Let's explore the steps required to solve this problem using the relevant kinematic equations.

Relevant Kinematic Equations

The following kinematic equations are essential for solving this problem:

s ut frac{1}{2}at^2

v u at

v^2 u^2 2as

Solving the Problem Using s ut frac{1}{2}at^2

Given:

Initial velocity (u) 0 m/s

Acceleration (a) 4 m/s^2

Distance (s) 200 m

The formula we need is:

s ut frac{1}{2}at^2

Substituting the values:

200 0 cdot t frac{1}{2} cdot 4 cdot t^2

This simplifies to:

200 2t^2

Dividing both sides by 2:

100 t^2

Solving for t:

t sqrt{100} 10 , text{seconds}

Hence, it will take the object 10 seconds to travel 200 meters at a constant acceleration of 4 m/s2 starting from rest.

Verification Using Other Equations

To further verify our solution, we can use the following derived values:

Velocity at the end of 200 meters:

v u at 0 4 cdot 10 40 , text{m/s}

Or, using the equation v^2 u^2 2as:

v^2 0^2 2 cdot 4 cdot 200 1600

v sqrt{1600} 40 , text{m/s}

As seen, the consistent results confirm the accuracy of our calculations.

Key Concepts and Practical Applications

This problem illustrates the practical application of kinematic equations in real-world scenarios. Understanding and applying these equations help in solving problems related to projectile motion, planetary movements, or any situation involving constant acceleration.

Conclusion

By applying the kinematic equations correctly, we can easily determine the time and velocity of an object moving under constant acceleration. This method is widely used in physics, engineering, and even everyday scenarios, demonstrating the importance of these fundamental mathematical tools.