Mastering kinematic equations: Understanding the basic principles of motion

Topics Covered In This Article:

• What is kinematics?
• Kinematic variables
• Kinematic equations
• Tips to solve kinematic equations
• Frequently asked questions (FAQ)

What Is Kinematics?

Kinematics is an integral part of physics that deals with the study of motion. It examines how objects move and change their positions over time, as well as the forces that cause these motions. This field of study looks at velocity, acceleration, displacement, and other related concepts to help explain how things move in the physical world. Learning about kinematics can provide a foundation for understanding more advanced topics in physics such as momentum, energy conservation, work-energy theorem and many more. Furthermore, kinematic equations are used to solve problems involving motion which makes it an important tool for problem-solving in O-level Physics exams.

Kinematic Variables

Before we get into the kinematic equations, let's learn about some important kinematic variables that are used in the kinematic equations.

1. Position: the location of an object in space, usually measured in meters \(\text{(m)}\).
2. Displacement: the change in position of an object, usually measured in meters \(\text{(m)}\).
3. Velocity: the rate at which an object's position changes. Velocity is usually measured in meters per second \(\text{(m/s)}\).
4. Speed: the rate at which an object covers distance, usually measured in meters per second \(\text{(m/s)}\).
5. Acceleration: the rate at which an object's velocity changes, usually measured in meters per second squared \(\mathrm{(m/s^2)}\).
6. Time: the duration of the motion or the time taken for an object to move from one position to another, usually measured in seconds \(\text{(s)}\).
7. Distance: the length of the path taken by an object during its motion. Distance is usually measured in meters \(\text{(m)}\).
8. Trajectory: the path followed by an object during its motion in space.
9. Momentum: the product of an object's mass and velocity, usually measured in kilograms meters per second \(\text{(kg·m/s)}\).

Kinematic Equations

Now that we know what is kinematics, and its different variables, let's learn about the equations that we can use to solve various problems of motion. For this article, we'll only focus on Kinematic equations for one-dimensional motion, also known as linear motion, and learn about the formulas, their significance in physics, and how they are applied in real-life situations.

What are kinematic equations?

Kinematic equations are mathematical equations that describe the motion of objects. They are derived from the equations of motion, which are a set of laws that describe how objects move.

These principles are used in real-life in the following situations:

• Motion in sports, such as football and baseball, to analyse trajectories and forces generated by players;
• Robotics for analysing the movements of robotic arms;
• Physics experiments to measure velocity or acceleration;
• Astronomy for calculating the orbits of planets around stars or asteroids around planets.

What are the five kinematic equations?

There are five kinematic equations that are commonly used in physics. They are:

• \(\Delta x={v_0}t + \frac{1}{2}at^2\)
• \(v_f = v_0 + at\)
• \(\Delta x = \frac{1}{2}(v_0 + v_f)t\)
• \(\Delta x = v_ft - \frac{1}{2}at^2\)
• \(v_f^2 = v_0^2 + 2a \Delta x\)

The five kinematic equations are a set of formulas used to describe the motion of an object in one dimension, also known as linear motion. Each equation relates four variables: displacement \((\Delta x)\), initial velocity \((v_0)\), final velocity \((v_f)\), acceleration \((a)\), and time \((t)\).

Let's understand each of these equations in detail.

1. The kinematic equation for displacement \((\Delta x={v_0}t + \frac{1}{2}at^2)\)

The equation \((\Delta x={v_0}t + \frac{1}{2}at^2)\) relates to the displacement \((\Delta x)\) of an object that is moving with a constant acceleration \((a)\) over a period of time \((t)\), starting with an initial velocity \((v_0)\).

The term \(v_0t\) represents the displacement that the object would have covered during the time \(t\), if it had moved at a constant velocity \(v_0\). The second term \(\frac{1}{2}at^2\) represents the additional displacement that the object covers due to its acceleration.

In simpler terms, the equation describes how the displacement of an object changes over time when it is subjected to a constant acceleration.

2. The kinematic equation for uniformly accelerated motion ​​\((v_f = v_0 + at)\)

The equation \((v_f = v_0 + at)\) is used to calculate the final velocity \((v_f)\) of an object in motion, given its initial velocity \((v_0)\), acceleration \((a)\) and the time (\(t\)) elapsed during the motion.

The equation is derived from the kinematic equations of motion, which describe the motion of objects in terms of their position, velocity, acceleration, and time. In this specific equation, the final velocity is equal to the initial velocity plus the product of acceleration and time.

For example, if an object is initially moving at a velocity of \(v_0 = 5 \text{ m/s}\), and experiences a constant acceleration of \(a = 2 \mathrm{\,m/s^2}\) for a time of \(t = 3\) seconds, we can use the equation \(v_f = v_0 + at\) to find its final velocity:

\(\begin{align}\begin{aligned} v_f &= v_0 + at \\[2ex] v_f &= 5 \text{ m/s} + (2 \text{ m/s}^2 \times 3 \text{ s}) \\[2ex] v_f &= 5 \text{ m/s} + 6 \text{ m/s}\\[2ex] v_f &= 11 \text{ m/s} \end{aligned} \end{align}\)

Therefore, the final velocity of the object after \(3\) seconds of acceleration is \(11 \text{ m/s}\).

3. The distance formula \((\Delta x = \frac{1}{2}(v_0 + v_f)t)\)

This equation \((\Delta x = \frac{1}{2}(v_0 + v_f)t)\) relates to the displacement \((\Delta x)\) of an object moving with a constant acceleration, given its initial velocity \((v_0)\), final velocity \((v_f)\) and time \((t)\) of motion.

The equation states that the displacement \((\Delta x)\) is equal to half the sum of the initial velocity \((v_0)\) and final velocity \((v_f)\), multiplied by the time \((t)\) taken for the motion.

In other words, the equation provides a way to calculate the change in position of an object when it starts with a certain velocity, undergoes acceleration and then ends with a different velocity. The equation assumes that the acceleration remains constant throughout the motion.

4. The displacement formula \((\Delta x = v_ft - \frac{1}{2}at^2)\)

The equation \((\Delta x = v_ft - \frac{1}{2}at^2)\) is a formula used to calculate the displacement or change in position \((\Delta x)\) of an object that is moving with a constant acceleration \((a)\) over a certain amount of time \((t)\).

The formula takes into account the initial velocity \((v_f)\) of the object, which is multiplied by the time interval \((t)\) to give the distance covered by the object in the absence of acceleration. The term \(\frac{1}{2}at^2\)takes into account the effect of acceleration on the object's position over time.

The factor of \(\frac{1}{2}\) represents half of the acceleration multiplied by the time interval squared, and it represents the change in position caused by acceleration.

Overall, the formula calculates the net displacement or change in position of the object by subtracting the effect of acceleration from the distance covered by the object in the absence of acceleration.

5. The kinematic equation for final velocity \((v_f^2 = v_0^2 + 2a \Delta x)\)

The equation \((v_f^2 = v_0^2 + 2a \Delta x)\) is a kinematic equation that relates the final velocity \((v_f)\) of an object to its initial velocity \((v_0)\), acceleration \((a)\) and displacement \((\Delta x)\).

The equation states that the final velocity squared \((v_f^2)\) is equal to the initial velocity squared \((v_0^2)\) plus two times the product of the acceleration \((a)\) and the displacement \((\Delta x)\) of the object.

In simpler terms, the equation means that the final velocity of an object depends on its initial velocity, how much it accelerates, and how far it travels. It can be used to calculate the final velocity of an object when its initial velocity, acceleration, and displacement are known.

Tips To Solve Kinematic Equations

1. Understand the concepts

Before you can start solving kinematic equation problems, it is important to have a solid understanding of the concepts involved. Make sure you understand the definitions of terms such as displacement, velocity, acceleration, and time. Knowing the basics is crucial for tackling more complex problems.

2. Draw diagrams

Once you have read the problem, make a rough sketch of the problem. This will help you understand the nuances of the question and understand which variables are at play.

3. Identify the variables

The first step in solving any kinematic equation problem is to identify the variables given in the problem. Look for information on displacement, velocity, acceleration, and time. Sometimes, you may need to use other formulas to find these variables before you can start solving the problem.

4. Choose the right formula

Once you have identified the variables, choose the right kinematic equation formula to use. Each equation is used to solve for a specific variable. Make sure you choose the equation that allows you to solve for the unknown variable.

5. Rearrange the equation

After choosing the equation, rearrange it to solve for the unknown variable. Make sure you isolate the variable you want to solve for on one side of the equation and everything else on the other side. This may require some algebraic manipulation, so make sure you have a solid understanding of algebraic concepts as well.

6. Plug in the values

Finally, plug in the values of the known variables into the equation and solve for the unknown variable. Make sure you use the correct units throughout the problem and round off your answer to the appropriate number of significant figures.

7. Practice, practice, practice

As with any skill, the key to mastering kinematic equations is practice. Solve as many problems as you can and make sure you understand the solutions. This will help you identify patterns and strategies that you can use to solve more complex problems.

By following these tips, you can become proficient in solving kinematic equation problems and score well on exams. Remember to always start by understanding the concepts, identify the variables, choose the right formula, rearrange the equation, plug in the values, and practice. With time and practice, you can master the art of kinematic equations and tackle even the most complex physics problems with ease.

Frequently Asked Questions (FAQs)

Q1. What are the three types of motion in kinematics?

The three types of motion in kinematics are linear, rotational, and oscillatory.

Q2. What is displacement?

Displacement is the change in the position of an object in one dimension. It is represented by the symbol Δx and is measured in meters.

Q3. What is the difference between distance and displacement?

Distance is the total length covered by an object during motion, while displacement is the change in position of the object from its initial to final position.

Q4. What is acceleration?

Acceleration is the rate at which an object's velocity changes with respect to time. It can be either positive or negative depending on whether an object is speeding up or slowing down. Acceleration can be found by calculating the change in velocity divided by elapsed time and is typically measured in meters per second squared (\(\text{m/s}^2\)). Kinematic equations such as Newton's Second Law are used to calculate acceleration when the force acting on an object and its mass are known.

Q5. What is the unit of acceleration?

The unit of acceleration is meters per second squared (\(\text{m/s}^2\)).

Q6. What is velocity?

Velocity is a measure of how fast an object is moving in a certain direction. It is defined as the rate of change of position over time, and is typically measured in meters per second (\(\text{m/s}^2\)). Velocity can be determined using kinematic equations such as the average velocity equation, which states that the average velocity of an object over a given time period can be calculated by taking the ratio of displacement divided by time.

Q7. Can velocity be negative?

Yes, velocity can be negative if the object is moving in the opposite direction of its initial motion.

Q8. What is the difference between speed and velocity?

Speed is the rate at which an object moves. On the other hand, velocity is the rate at which an object moves in a specific direction.

Q9. What is the formula for average speed?

The formula for average speed is the total distance travelled divided by the total time taken.

Q9. What is the difference between instantaneous and average velocity?

Instantaneous velocity is the velocity of an object at a specific instant in time, while average velocity is the total displacement of the object divided by the total time taken.

Q10. What is the formula for acceleration?

The formula for acceleration is the change in velocity divided by the change in time.

Q11. What is the difference between positive and negative acceleration?

Positive acceleration occurs when the velocity of an object increases in the direction of its initial motion, while negative acceleration occurs when the velocity of an object decreases in the direction of its initial motion.

Q12. What is the difference between uniform and non-uniform acceleration?

Uniform acceleration occurs when the acceleration of an object remains constant over time, while non-uniform acceleration occurs when the acceleration of an object changes over time.

Q13. What is force?

Force is an interaction between two objects that causes one to accelerate relative to the other. It is typically caused by contact or gravitational attraction, and it is measured in Newtons (\(\text{N}\)). In order to understand how forces affect motion, one needs to understand concepts such as momentum, impulse and work. Kinematic equations are often used to calculate force based on these concepts.

Q14. What is energy?

Energy refers to the capacity for doing work, and it comes in many forms including potential energy, kinetic energy and thermal energy. Potential energy is stored energy due to an object’s position relative to other objects while kinetic energy reflects the motion of an object through space. Thermal energy refers to the heat generated from chemical reactions or friction between moving objects. Kinematic equations such as the Work-Energy Theorem allow us to calculate both kinetic and potential energies associated with a given system of objects.

Q15. What is momentum?

Momentum is defined as mass times velocity, and it determines how much “punch” a moving body has against another body when they collide or interact with each other. Momentum remains constant unless external forces act upon it, meaning that if two objects have equal masses but different velocities then they will have different momentums; similarly, if two objects have equal velocities but different masses then their momentums will also be different. Kinematic equations such as conservation of momentum can therefore help us understand interactions between moving bodies, allowing us to predict their behaviour after impacts or collisions occur.

Q16. How are kinematic formulas applied in physics?

Kinematic formulas are used to solve problems involving one-dimensional motion. These problems may include finding the displacement, velocity, acceleration, or time of an object.

Q17. What are some common misconceptions about kinematic formulas?

One common misconception is that kinematic formulas only apply to objects moving in a straight line. However, they can also be applied to objects moving in a curved path.

Another misconception is that kinematic formulas can only be used to solve problems involving constant acceleration. However, they can also be used to solve problems involving non-uniform acceleration.

Q18. How can I use kinematic formulas to solve problems?

To use kinematic formulas to solve problems, you need to identify the variables that are given and the ones that are unknown. Then, you can use the appropriate formula to solve for the unknown variable.

Geniebook's science teachers have prepared a set of practice questions for those interested in solving problems related to kinematic equations. You can access the questions and hone your skills by clicking here. 

Understanding kinematic equations is crucial for anyone studying physics or interested in understanding the basic principles of motion. By following the tips outlined in this article and practising with examples, you can become proficient in solving problems involving kinematic equations. Remember to always identify the variables given in the problem, use the appropriate equation, rearrange the equation to solve for the unknown variable, and plug in the values of the known variables. With practice, you can master the art of kinematic equations and tackle even the most complex physics problems with ease.