Equations of motion - derivation and applications

Kinematics and dynamics are important parts of physics that deal with the motion of objects. In this article, we will explore the equations of motion that are used to describe the motion of objects. These equations are essential tools in physics and engineering, helping us predict the future position and velocity of moving objects.

Understanding momentum

Motion is the change in the position of an object over time. To understand motion, we need to use quantities like distance, displacement, speed, velocity and acceleration. Let us briefly describe these terms:

• Distance is the total path length covered by an object, regardless of its initial or final point. It is a scalar quantity.
• Displacement is the shortest distance from the initial position to the final position of an object. It has both magnitude and direction, which makes it a vector quantity.
• Speed is the rate at which an object travels a distance. It is the magnitude of velocity and is scalar.
• Velocity is the rate of change of displacement. It is a vector quantity and includes direction.
• Acceleration is the rate of change of velocity. It shows how quickly the velocity changes.

Introduction to equations of motion

The equations of motion describe the relationship between these quantities for objects moving with constant acceleration. They help us solve problems involving the motion of objects. The three elementary equations of motion, also called the "linear" equations (because they involve displacement s, initial velocity u, final velocity v, acceleration a and time t), are:

Derivation of equations of motion

First equation of motion

The first equation relates final velocity (v) to initial velocity (u), acceleration (a) and time (t). The equation is derived from the definition of acceleration, which is:

a = (v - u) / t

Rearranging this equation, we get:

v = u + at

This equation tells us how the velocity of an object with constant acceleration changes with time.

Second equation of motion

The second equation gives the displacement (s) of an object in terms of initial velocity, time, and acceleration. The average velocity of an object moving with constant acceleration is the average of its initial and final velocities:

v_avg = (u + v) / 2

Displacement can be expressed as:

s = v_avg × t

Substituting v from the first equation:

s = ((u + (u + at)) / 2) × t

On simplifying, we get:

s = ut + (1/2)at 2

Third equation of motion

The third equation of motion relates the square of the final velocity to the square of the initial velocity, acceleration, and displacement. Starting with the first equation, let's solve for time:

t = (v - u) / a

Substitute this value of t into the second equation:

s = u((v - u) / a) + (1/2)a((v - u) / a) 2

Simplify:

v 2 = u 2 + 2as

Visual example

In the figure above, a red ball starts at an initial position u and moves towards a blue ball at a final position v. The line represents the displacement s.

Applications of equations of motion

Equations of motion are widely used in various fields such as engineering, aerospace, automotive, and even sports. Here are some examples:

1. Design of transportation systems

Engineers use these equations to design transportation systems, calculating how long a train should stop or how fast a car should accelerate to get onto the highway.

Example: If a car accelerates from rest at a rate of 3 m/s 2 for 10 seconds, what will be its final velocity?

v = u + at = 0 + (3 × 10) = 30 m/s

2. Aerospace engineering

These equations help understand projectile motion and are important in sending spacecraft into orbit. The calculations involve determining the speed and position of satellites and other space objects.

Example: A rocket moves upward with a uniform acceleration of 5 m/s 2 Starting from rest, calculate the displacement traveled by the rocket in 8 seconds.

s = ut + (1/2)at 2 = 0 × 8 + (1/2) × 5 × (8) 2 = 160 m

3. Sports analysis

Understanding motion helps athletes improve performance by analyzing the dynamics of their movements. In sports such as javelin throw or shot put, knowing the equations helps maximize the distance thrown.

Example: A javelin is thrown with an initial speed of 20 m/s at an angle where the final velocity becomes 0 m/s before it hits the ground. Find the total time of flight if the acceleration in the air is -9.8 m/s 2 (assuming vertical motion only).

v = u + at => 0 = 20 - 9.8t => t = 20 / 9.8 ≈ 2.04 seconds

Conclusion

The equations of motion provide a powerful tool for solving problems involving the motion of objects under constant acceleration. They simplify the description of motion and help us predict the future motion of objects. Understanding these equations is fundamental for students and professionals in many science and engineering fields.

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