## What are Kinematic Equations in Physics?

Kinematic equations of motion in physics are a set of equations that describe the motion of objects in terms of position, velocity, acceleration, and time. These equations are derived from the three fundamental kinematic equations:

**First equation**

v = u + at

**Second equation**

s = ut + (1/2)at^{2}

**Third equation**

v^{2} = u^{2} + 2as

**Fourth Equation**

s = [ (u + v) / 2 ] x t

Where:

- v is the final velocity
- u is the initial velocity
- a is the acceleration
- s is the displacement
- t is the time

First kinematic Equation | v = u + at |

Second kinematic Equation | s = ut + (1/2)at^{2} |

Third kinematic Equation | v^{2} = u^{2} + 2as |

The above equations are only valid for motion undergoing uniform acceleration. Hence, the acceleration remains constant throughout the motion. For non-uniformly accelerated motion, the equations of motion are more complex.

## Explanation of Kinematic Equations

If you’ve ever played a sport, taken a car ride, or simply walked down the street, you’ve experienced motion. But have you ever stopped to consider the underlying principles of motion? **Kinematics is an arm of physics that studies motion. On the other hand, kinematics equations are the mathematical formulas we use to describe and predict the behavior of moving objects**.

At its most basic level, kinematics is the study of motion without considering the forces that cause it. Additionally, we also refer to kinematics equations as the **equations of motion**. These are mathematical formulas that describe the relationship between an object’s position, velocity, and acceleration over time. Therefore, these equations are fundamental to understanding and predicting the behavior of moving objects. In this article, we’ll explore the basics of kinematics equations, how they work, and why they’re important.

## Displacement, Velocity, and Acceleration

To understand kinematics equations, it’s important to first understand the concepts of displacement, velocity, and acceleration. **Displacement** is the change in an object’s position from its initial location to its final location. **Velocity** is the rate at which an object changes its position.** Acceleration** is the rate at which an object changes its velocity.

## How to Derive Kinematic Equations

The derivation of kinematics equations is based on the following assumptions:

- The object is moving in a straight line.
- The acceleration is constant.
- The velocity and acceleration are in the same direction.

Using these assumptions, we can derive the kinematic equations.

### Deriving First Equation of Motion

We can derive the first equation by using the definition of acceleration, which is the rate of change of velocity with time. Therefore, we can now write:

Acceleration, a = change in velocity (v – u) / time (t)

Which will become

a = (v – u) / t **[Remember that acceleration, a = change in velocity / time]**

After rearranging the above equation and making v subject of the formula, we will obtain:

**v = u + at** **[This is the first equation of motion]**

### Deriving the Second Equation of Motion

To obtain the second equation, we will apply the definition of displacement. Since we know that displacement is the change in the position of an object. Similarly, we can define displacement as a measure of the difference between two points in a specific direction. Consequently, the above statement will lead us to the equation below

Since average velocity, v_{a} = (u + v) / 2

Hence, the distance covered will become

s = [(u + v) / 2] t [Because v = s/t which shows that s = vt]

We can insert **v = u + at** into the above equation to obtain

s = [(u + v) / 2] t = [(u + u + at) / 2] t

Thus,

We will have s = [(2u + at)/2]t = ut + (1/2)at^{2}

**Therefore, our second equation of motion will be **

**s = ut + (1/2)at ^{2} **

### Deriving the Third Equation of Motion

The third equation is derived by using the first two equations and eliminating time. We can write:

v = u + at

Since s = ut + (1/2)at^{2}

Hence,

By squaring both sides of **v = u + at** we will end up with

v^{2} = (u + at)(u + at) = u^{2} + 2uat + a^{2}t^{2}

Therefore,

v^{2} = u^{2} + 2a[ut + (1/2)at^{2}]

and ut + (1/2)at^{2} is equal to s

**Therefore, our third equation of motion will become**

**v ^{2} = u^{2} + 2as**

## Applications of Kinematics Equations

Kinematics equations are used in various fields, including physics, engineering, and robotics. Some of the applications of kinematics equations are:

- Calculating the trajectory of a projectile: Kinematics equations can be used to calculate the path of a projectile, such as a cannonball or a rocket, under the influence of gravity.
- Designing roller coasters: Kinematics equations can be used to design roller coasters by calculating

## Solved Problems Using Kinematic Equations

Here are some solved problems to help you understand how to apply kinematic equations (equations of motion):

### Problem 1

A bus traveling at 60 km/h accelerates uniformly at 5 m/s^{2}. Calculate its velocity after 2 minutes.

#### Answer

**The final answer to this question is 617 m/s or 2221 km/h**

#### Explanation

**Data:**

Initial velocity, u = 60 km/h = (60 x 1000) / 60 x 60 = 16.6 m/s = 17 m/s

Acceleration, a = 5 m/s^{2}

Time, t = 2 min = 2 x 60 = 120 seconds

**Unknown**

Final velocity, v = ?

**Formula**

We will apply the first equation of motion **v = u + at**

#### Solution

We will insert our data into the formula

**v = u + at = 17 + (5 x 120) = 617 m/s**

We can convert the above answer into km/h as

v = (617 / 1000) x 60 x 60 = 2221.2 km/h = 2221 km/h

### Problem 2

A car accelerates uniformly at a rate of 10 m/s^{2} from an initial velocity of 36 km/h for 30 seconds. Find the distance covered during this period.

#### Answer

**The final answer to this question is 4.8 km**

#### Explanation

**Data**

Initial velocity, u = 36 km/h = 10 m/s

Acceleration, a = 10 m/s^{2}

Time, t = 30 seconds

**Unknown**

Distance covered, s = ?

**Formula**

We will use the second kinematic equation which says

s = ut + (1/2)at^{2}

#### Solution

We will have

**s = ut + (1/2)at ^{2} = 10 x 30 + 0.5 x 10 x 30^{2} = 300 + 4500 = 4800 m **

We can convert the above answer (4800 meters) into km by saying

**s = 4800 / 1000 = 4.8 km**

### Problem 3

A body moving with an initial velocity of 30 m/s accelerates uniformly at a rate of 10 m/s^{2} until it attains a velocity of 50 m/s. What is the distance covered during this period?

#### Answer

**The final answer to this question is 80 meters**

#### Explanation

**Data**

Initial velocity, u = 30 m/s

Acceleration, a = 10 m/s^{2}

Final velocity, v = 50 m/s

**Unknown**

Distance covered, s = ?

**Formula**

From the third kinematic equation

v^{2} = u^{2} + 2as

and then make s subject of the formula to obtain the equation

s = (v^{2} – u^{2}) / 2a

We will use the above equation to find the distance covered

#### Solution

By putting our data into the formula, we will have

s = (v^{2} – u^{2}) / 2a = (50^{2} – 30^{2}) / 2 x 10

Which implies that

**s = 1600 / 20 = 80 m**

## Frequently Asked Questions (FAQs)

**What is the kinematics equation?**

Kinematics equations are a set of equations that describe the motion of objects in terms of their position, velocity, and acceleration.

**What are the three kinematics equations?**

The three kinematic equations are:

**1)** v = u + at

2) s = ut + (1/2)at^{2}

3) v^{2} = u^{2} + 2as

**How do you solve kinematic equations?**

To solve kinematic equations, you need to identify which equation(s) to use, plug in the known values, and solve for the unknown value(s) using algebra.

**What is the kinematic formula for distance?**

The kinematic formula for distance is s = ut + (1/2)at^{2}, where s is distance, u is initial velocity, t is time, and a is acceleration.

**What is the kinematic equation for velocity?**

The kinematic equation for velocity is v = u + at, where v is final velocity, u is initial velocity, a is acceleration, and t is time.

**What is the kinematic formula for acceleration?**

The kinematic formula for acceleration is a = (v – u)/t, where a is acceleration, v is final velocity, vi is initial velocity, and t is time.

**What is the difference between kinematics and dynamics?**

Kinematics is the study of motion without considering the forces that cause the motion, while dynamics is the study of motion taking into account the forces that cause the motion.

**What is kinematics used for?**

Kinematics is used in many fields, including physics, engineering, and robotics, to describe and analyze the motion of objects and systems.

## Test Your Understanding

**Question: **What is the kinematic equation for time?**Answer:** There is no specific kinematic equation for time. Time is usually given in the kinematic equations as a known value that is used to solve for other variables such as distance, velocity, or acceleration.

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