Kinematic equations for uniformly accelerated motion

Introduction

Classical physics considers the movement of any body with or without consideration of the forces applied on it. This particular branch of physics is divided into several types of subgroups, out of which kinematics equations and the translation movement of objects are considered. When an object moves in a straight line, there are several things one needs to consider, starting from instantaneous velocity to uniform acceleration. In this post, we will cover all the basic terminologies related to motion and also the three motion equations, their graphs, and derivations.

Introduction to Kinematics and basic terminologies

For understanding the kinematics formulas, there are some preliminary topics that need to be understood properly. In the below section, we will describe them first so that you can have more clarification in solving problems using the kinematics equations.

Rest and motion

The first thing you need to know is rest and motion. When an object is static in a frame of reference and does not displace from its initial position, the state is known as rest in that frame of reference. On the contrary, the state of motion is defined when an object displaces from one position to another with respect to a frame of reference.

Let’s take an example of a passenger sitting on the train and another person standing on the platform. When the train starts to move, the person standing on the platform will consider the train to be in motion. On the other hand, the person sitting inside will consider himself to be at rest with respect to the train’s ceiling and floor.

Thus rest and motion are frame dependent.

Point object

To explain the kinematic equation for uniformly accelerated motion, a point object is always considered. If we consider any normal object, it can be in motion and therefore, it will vibrate. But, when we consider the object to be a particle, the vibrational and translational motion can be ignored easily. This object can be considered as a point object having no dimension or extent.

What is a cartesian coordinate system?

In classical mechanics, the cartesian coordinate system plays a very crucial role. In this system, we consider three axes: the X-axis, Y-axis, and Z-axis. When we consider a frame of reference, we need to include two axes together, and they will be 90-degrees apart from each other. For example, in the XY plane, the total angular value is 90-degrees, while the same is applicable for the YZ plane and XZ plane.

When a point object is considered, it needs to be represented on any one of the planes, and the position will be determined by two coordinates of the respective axes. For example, if a point object O has a position of (2, 3) in the XY plane, it means that O is located at 2 units along the X-axis and 3 units along the Y-axis from the origin (0, 0).

Types of motion

Based on the cartesian system, there are three types of motion.

1. One-dimensional motion that occurs along a single axis, X, Y, or Z.
2. Two-dimensional motion that happens along a plane like XY, YZ, and ZX.

iii. Three-dimensional motion, which happens in most cases involving coordinates of all the three axes.

Even though there are three main motion types, in the kinematics equations, we consider the single dimension motion only. For two-dimensional motions, projectile motion is the best example.

Terminologies for the motion of an object

There are several terminologies that need to be learned to explain the kinematic equation for uniformly accelerated motion. These are:

Distance and displacement

Distance is termed as the total length travelled by a body, while displacement can be termed as the shortest distance between the initial and final point. Both the parameters have SI units as metre or m.

Speed and velocity

Speed is defined as the total distance covered in a given time while velocity is the rate of change of displacement.

Speed = Distance/time

Velocity = Displacement/ time

Or, δv = δs/δt …. (i)

Or, v = s/t …. (ii)

Equation (i) represents the value of instantaneous velocity, while equation (ii) is for linear uniform velocity. The SI unit of velocity is ms-1.

Acceleration

The rate of change of velocity is known as acceleration. It is described by the formula of:

δa =δv/δt …… (i)

Or, a = v/t (ii)

Equation (i) represents the value of instantaneous acceleration, while equation (ii) is for linear uniform acceleration. The SI unit of acceleration is ms-2.

Equations of kinematics concerning uniformly accelerated motion

There are three kinematic equations that define the linear motion of an object along a single axis with uniform acceleration. Here, both initial and final velocities are considered. To understand the equations, the following variables are involved:

• S = displacement of the body
• u = initial velocity of the body
• v = final velocity of the body
• t = time taken to travel by the body
• a = uniform acceleration of the body

For linear motion with positive acceleration, the three kinematics formulas are given as follows:

v = u + at ……. (1)

S = ut + 1/2at2 ……. (2)

v2 = u2 + 2aS ……. (3)

When deceleration or retardation is considered, these three kinematic equations can be written as:

v = u – at ……. (4)

S = ut- 1/2at2 ……. (5)

v2= u2- 2aS ……. (6)

Note: In equations (4), (5) and (6) a is considered as retardation and hence negative need not be taken again.

Equations of free-fall motion under gravity

When a body freely falls from a height, its motion is controlled by the acceleration due to gravity or g, whose value is considered as 9.8 ms⁻² on earth. In such cases, displacement S is replaced by height h in the kinematics formulas.

v = u + gt ……. (7)

h = ut + 1/2gt2 ……. (8)

v2 = u2 + 2gh ……. (9)

If the body moves against the acceleration due to gravity, like a ball being thrown in the upward direction or someone climbing a ladder, to explain the kinematic equation for uniformly accelerated motion, the (-g) is considered in the form of (+g).

Conclusion

In this post, you have learned everything related to basic kinematic equations, rest, motion, point objects, cartesian coordinates, and more. However, this is based solely on linear motion, where the body is travelling straight along a line. These equations will automatically change when we consider the curvilinear motions or rotational motions. Also, the behaviour of any body based on kinematics can be understood well through numerical examples because then you can implement the kinematic formulas easily.