The Four Basic Equations in Kinematics (Linear Motion)

Why is there always something wrong with my equations in Physics, Mechanics, or Kinematics? — Have you ever thought about this? Then this blog account is for you. It will explain — in detail — everything you are doing wrong. If you want to make sure you’re not doing any silly mistakes, follow me at MechanicallyChallenged on medium.com

Let’s start with the most basic equations these subjects have to offer.

These are The Four Basic Kinematics equations, which are used a lot, so if you don’t understand what these actually do and the assumptions you take, then this will cause you plenty of headaches.

The First Mistake

Why are there two ways of writing it?

They are both valid, however, the one on the left is more accurate. This is where most people make the first mistake. Displacement and Distance are two different things. Displacement \Delta X is what these equations measure. They are NOT measuring Distance s.\Delta x is more descriptive of what we’re actually doing, which is measuring the change in position.

Displacement

• Vector
• The RIGHT one to use in the equations
• Measures the difference in the start position and end position

Distance

• Scalar
• The WRONG one to use in the equations
• Measures the entire stretch traveled from the start position to the end position

If you’re still unsure about the difference between distance and displacement, then there are plenty of videos explaining it well, for example, this one:

The Second Mistake

Some people use Velocity and Speed interchangeably, but they are not the same. This is similar to how people mistake displacement and distance.

Velocity

• Vector
• The RIGHT one to use in the equations
• Measures the difference in the speed at the start point and the speed at the endpoint

Speed

• Scalar
• The WRONG one to use in the equations
• Measures the speed needed to cover a distance over a certain time.

Again, if you’re still not sure about the difference, there are plenty of good videos out there explaining the difference, like this one:

https://www.youtube.com/watch?v=ukWpTQq13dw

The Assumptions Mistake

When using these equations, we are making big assumptions that will haunt us if we’re not aware of them.

These ONLY WORK for Constant Acceleration. Does that mean we throw these equations in the garbage when looking at a problem with varying acceleration? — No!

Calculating with differing acceleration

We look at the origin of the equations. In line 1. we can see how displacement, velocity, and acceleration relate to each other. The acceleration is the derivative of velocity vas a function and the double derivative of displacement \Delta x as a function. If we wanted to go in the opposite direction, we could get the velocity function by integrating the acceleration function and we could get the displacement function by integrating the velocity function.

In conclusion, if the acceleration is given as a function (not constant), then we need to use integration and the relationship between displacement, velocity, and acceleration to find a new equation that makes either displacement or velocity an equation in the form of a function that is dependent on time. Or velocity dependent on displacement, which is the case in the last line.

Do you recognize the equations above, and do you see which one is missing?

This one is missing. Why is that? There is no acceleration! The trained eye can also see that this equation is

displacement = avg. velocity * t

Since both the average velocity is constant, and time is constant, we’re going to have a hard time converting this into a function. In fact, it’s impossible. Although, this formula is constant, do not confuse it with the formula

stretch=speed*time

The formula used in Physics/Kinematics is the one with displacement and average speed because we’re often interested in the direction and magnitude when solving real-life problems, hence we use vectors. The other one consists of scalars and might be introduced in Elementary school to do simple calculations.

Example:

These formulas would quickly differ if you measured the time a car took to travel from one city to another. The formula using displacement would give the answer for the straight line going from city to city, while the formula using stretch would give the answer for how much time the entire path with turns and all took.

In some cases, they can be used interchangeably, depending on what problem you are solving. However, it’s good practice to use the right formula for the right thing.

Hope this helps!

Many of the same rules apply to the equivalent equations in Rotational Motion, which we will be looking closely at in the next post.