# Motion in a Plane

## Motion in a Plane Introduction

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 the 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 and the product so 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, so 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:

ax  = $\frac{dvx}{dt}$ ….(2)

ay  = $\frac{dvy}{dt}$ ….(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 + ax

s = uxt + ½ axt²

vx² = 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 + ay

s = uyt + ½ ayt²

vy² = uy² + 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:

The equation for the body executing a projectile motion is:

y = ax + bx² …..(7)

Question 1: What is Motion in a Plane Class 11?

Answer: Motion in a Plane Class 11 comes under the subject Physics Chapter 4. Motion in a plane is called a motion in two dimensions. Its examples are projectile motion, circular motion. For the analysis of such motion, our reference is an origin and two co-ordinate axes viz: X and Y-axis and scalar and vector physical quantities.

Question 2: What is an Example of Motion in a Plane?

Answer: The motion of an object in a plane gives us equations of its motion along the axes or two dimensions. For the X-axis: One of the most common examples of motion in a plane is Projectile motion. In a projectile motion, the only acceleration acts on the object in the vertical direction which is actually the acceleration due to gravity (g), i.e., 9.8 ms⁻².

Question 3: How Do We Study the Motion in a Plane?

Answer: We study the motion in a plane with the help of topics like vectors, projectile motion, displacement, acceleration, time, relative motion, and many more physical quantities. In a plane, we apply the same equations of motion separately in both directions: X -xis and Y-axis.

Question 4: What is Two-dimensional Projectile Motion?

Answer: In a 2-D projectile motion, throwing a rock or kicking a football ball generally produces a projectile pattern of motion that has both a vertical and a horizontal component.

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