Components Of Velocity Of Projectile

The projectile motion refers to a motion where the movement of an object takes place in a bilaterally symmetrical manner. Such a path is parabolic in nature, and it is known as trajectory. Projectile motion takes place when there is an application of one force on a projectile. This happens at the trajectory’s beginning, after which gravity is the only interfering force. This motion gives rise to certain components of velocity, known simply as the components of velocity of a projectile. 

Understanding Projectile and Projectile Motion

Before learning about the components of velocity of a projectile, we must understand the basics of a projectile. A projectile is an item upon which gravity force is the only force that influences it. You can find plenty of examples of projectiles in existence in real life. An object that is thrown from the building’s top is a projectile. However, the air resistance influencing it, in this case, should be negligible. Another example can be an object that is thrown in a vertical, upward manner. However, air resistance should not affect it in any way.

Projectiles are those objects that continue in motion by their own inertia. This takes place after they are projected into the air. Only gravity is the force that impacts it after its projection. 

The projectile motion refers to an item’s motion, whose projection takes place into the air. After the initial launching force, the object only undergoes the force of gravity. The path that this object follows is known in the field of physics as trajectory. While travelling through the air, a frictional force will act on the object because of air resistance. 

Horizontal and Vertical Components of the velocity of a projectile

The Horizontal and Vertical Components of the velocity of a projectile are as follows:

Horizontal distance

Horizontal distance = (initial horizontal velocity)(time)

x = vxo t

Vertical distance

Vertical distance = (Initial vertical velocity)(time) – ½(acceleration gravity)(time)2

Horizontal velocity

Horizontal velocity = initial horizontal velocity

vx = vxo

Vertical velocity

Vertical velocity = initial vertical velocity – (gravity acceleration)(time)

vy = vyo – gt

vy = vyo – (9.80 m/s2)(t)

So, vertical velocity = vyo – (9.80 m/s2)(t)

Here, x = horizontal distance (m)

y = vertical distance (m)

Also, v = velocity (combined components, m/s)

Moving on,  

vxo = initial horizontal velocity (m/s)

vyo = initial vertical velocity (m/s)

t = time (s)

Also, g = 9.80 m/s2

This is the acceleration due to gravity.

Initial Velocity

The expression of the initial velocity can take place as:

• x components 
• y components

ux = u⋅cosθ 

uy = u⋅sinθ

In this equation, u represents initial velocity magnitude while a projectile angle is represented by θ.

Time of Flight

This refers to the time taken from the projection of an object till it reaches the surface. This time is dependent on the following factors:

The initial velocity magnitude

The angle of the projectile

T = 2⋅uy/g 

T = 2⋅u⋅sinθ/g

Acceleration

In projectile motion, acceleration does not take place in the horizontal direction. The acceleration, a, happens only in the vertical direction. The main and only factor behind it is gravity. This can be expressed as:

ax = 0

ay = −g

Velocity

The horizontal velocity is characterised in projectile motion by remaining constant. However, there is linear variation here due to constant acceleration. At any time, the velocity is:

ux = u⋅cosθ

uy = u⋅sinθ−g⋅t

Use of Pythagorean Theorem can also take place to find it.

Displacement

At time, t, the displacement components of velocity of projectile are:

x = u⋅t⋅cosθ

y = u⋅t⋅sinθ−1/2gt2

Parabolic Trajectory

The displacement equations can be used for the measurement of the parabolic form of a projectile motion equation:

Y = tanθ⋅x – g.x2/2.u2.cos2 θ

Maximum Height

Maximum height is determined when vy = 0. Using this, a rearrangement of the velocity equation can take place to ascertain the time it will take for acquiring the maximum height

Th = u⋅sinθ/g

Here, Th represents the time for acquiring the maximum height. From the displacement equation, it is possible to calculate the maximum height as follows:

H = u2⋅sin2θ/2⋅g

Range

The fixation of the motion’s range takes place by the condition y = 0. Using this, a rearrangement of the parabolic motion equation can take place for finding out the motion’s range:

R = u2⋅sin2θ/g

Conclusion

The projectile motion is a motion where an object’s movement takes place in a bilaterally symmetrical manner. Such a path is parabolic and is called a trajectory. The projectile motion gives rise to certain components known as components of velocity of a projectile.