Motion in a Plane

A body moving from one point to different points on the X and Y-axis is said to be executing motion in a plane. A plane comprises the X and Y-axis on which if we make the distance at the X-axis and the time at which a body moves along the vertical or Y-axis, then dividing the distance covered by the time taken we get the velocity. 

Similarly, on plotting the velocity along the X-axis and the time at the Y-axis, the product obtained is the acceleration of the body. 

Here, we will discuss all motion in a plane with its detailed introduction and formulas.

Parameters of Motion in a Plane

In the above heading, we discussed three parameters of motion, viz: distance, velocity, acceleration; besides these three, we have a displacement as well. Now, let’s introduce the concept of a motion in a plane in detail:

For understanding motion in a plane, it is necessary to understand motion in one dimension and the following parameters of motion in detail.

Distance: 

It is an overall measurement of the body that is calculated from the point an object initiates its journey to the point it terminates its journey. It is a scalar physical quantity, so we won’t be sure of in which direction we are travelling along with the train, we just know the distance we covered from Delhi to Bangalore.

Time: 

We are moving along with the time, so a factor through which we can determine the velocity and acceleration of an object is the time; however, it is a scalar quantity, so we just know the time we would reach Delhi from Dehradun, not the direction the train takes.

Velocity: 

It is a physical quantity that describes the magnitude and direction of a moving object. A velocity demonstrates how an object can be defined as the rate of change of the object’s position with respect to a frame of reference and time. Well! It might sound complicated because velocity is basically the speed of an object in a specific direction. 

Displacement: 

It is also a physical quantity that describes both the magnitude and direction of a body executing motion; however, it is the shortest distance a body can take reaching from one point to another.

Motion in a Plane 

We already know that velocity is a vector quantity, and therefore, by Pythagoras theorem, the magnitude of the velocity vector is given by:

\[|v| = v = \sqrt{vx^{2} + vy^{2}}\]...........(1)

Since we are considering motion in a plane, we determined the velocity along both the axis and then calculated the magnitude of a velocity vector by applying the Pythagoras theorem. 

For acceleration along both the axis, we have the following two equations:

\[a_{x}  = \frac{dvx}{dt}\].....….(2)

\[a_{y}  = \frac{dvy}{dt}\].......….(3)

Motion in Plane Equations

v= u + at…..(4)

s = ut + ½ at² …..(5)

v² = u² + 2as …..(6)

Here, equations (4), (5), and (6) are motion in a plane formulas for a particle ‘P’ executing motion in a plane, let’s define these one-by-one:

u = initial velocity 

v = final velocity

s = displacement of particle ‘P’

t = time the particle takes while executing a motion

a = acceleration of the particle executing motion in a plane

For a particle moving along the X and Y-axis, the above equations: (4), (5), (6) becomes in the following manner:

For X-axis:

vx   = u + axt 

s = u_xt + ½ axt²

v_x² = ux² + 2axs

The definition also changes in the following way:

u = initial velocity along the X-axis 

v = final velocity along the X-axis

s = displacement of particle ‘P’ along the X-axis

t = time the particle takes while executing a motion along the X-axis

a = acceleration of the particle executing motion in a plane along the X-axis

Now, for Y-axis:

vy   = u + ayt 

s = u_yt + ½ ayt²

v_y² = u_y² + 2ays

The definition also changes in the following way:

u = initial velocity along the Y-axis 

v = final velocity along the Y-axis

s = displacement of particle ‘P’ along the Y-axis

t = time the particle takes while executing a motion along the Y-axis

a = acceleration of the particle executing motion in a plane along the Y-axis

Now, let’s see some examples of real-life objects making a motion in a plane:

Examples of 2-D Motion in a Plane

• Throwing a cricket ball or a cannonball.

• The motion of a billiard ball along with the floor of the billiard table.

• A downstream or upstream motion of a boat in a river.

• A circular motion/revolution of the Earth around the Sun.

• A projectile motion of a bullet fired from a gun

Projectile motion is one of the best examples of an object bearing motion in a plane; let’s discuss it:

(Image will be uploaded soon)

The equation for the body executing a projectile motion is:

y = ax + bx² …..(7)

Best Way to Learn the Equations of Motion

The best way to understand any concept of physics is to grasp the significance of the variables of a particular formula. Solving equations of motion is very easy since there are only three formulas that the students have to remember in order to solve the equations. The students need to learn and understand the concept of motion at first and then they should go through the elaborate examples included in their textbooks so that they can get a clear idea of what this particular law signifies.

The best way to solve the equation is to understand the relationship between the variables in a formula. They need to remember what the variables signify and how the velocity and the other variables can influence the motion of the particular object. There are various examples included in their NCERT textbooks that show how to solve equations of motion. The students need to go through all the examples carefully and understand the pattern that they have to follow in order to solve.

Another way is to write down the formula and the concept along with the necessary information as soon as the students get to learn about them. Writing down any information will help the students to retain the piece of information for a longer period of time. In order to boost their self-confidence and avoid any kind of error in solving the equations, they need to practice and verify whether they are following the correct pattern to solve them or not.