Atlearner-
April 19, 2023

October 10, 2021

If an object is thrown horizontally without being thrown vertically upwards, the object no longer returns to the point of projection, rather than it moves slightly away from the point of projection.

It is actually thrown this way to move the object a little farther horizontally. For example, kicking a football. The object that is thrown here is called a **projectile** and the path followed by the projectile is known as **trajectory**.

Observing the Projectile Motion we can determine:

1. Equation of path of a projectile

2. Time of flight of a projectile *(Tf)*

3. Maximum height of a projectile *(Hm)*

4. Horizontal range of a projectile *(R)*

In the above projectile motion, the point O is called the **point of projection**, θ is the **angle of projection**, and OB = **horizontal range**. The total time taken by the object from reaching O to B is called the **time of flight**.

Now we can use the equations of motions to find out different parameters related to projectile motion.

In terms of components:

For the 2-dimensional motion the formula for the position vector 'r' ( Read more from what is position vector and displacement vector?)

In terms of components:

Here, *u = initial velocity, v = final velocity, a = acceleration due to gravity, r = position vector, t = time*.

In a Projectile Motion, there are two independent components of rectilinear motions:

The components of the initial velocity *(ux) = ucos𝜃*

the component of acceleration *(ax) = o*

The components of the initial velocity *(uy) = usin𝜃*

The component of acceleration *(ay) = -g*

When a particle is projected into the air at some speed, the only force acting on it during its flight in the air is gravity (g). This acceleration only works vertically downwards.

There is no acceleration in the horizontal direction. So the velocity of the particles in the horizontal direction remains constant.

Now if we take the initial position of the object to be the origin of the reference frame, we have: *x0 = 0*, *y0 = 0*

Then from equation no-(5, 6), we get

The components of velocity at time t can be obtained using equation no-(2, 3)

What type of trajectory a projectile followed can be seen through expressions of *x* and *y* as given above.

This is the Equation of path/trajectory of a projectile motion. By comparing this equation with *y = a x + b x2*, (Here a and b are constants) we can say that the path of a projectile is parabolic in nature.

How much time does a projectile take to reach the maximum height can be determined through expressions of *v**x* and *v**y* as given above.

Here *t**m* is the time taken by a projectile to reach the maximum height where *v**y = 0*

Now if we take the *T**f* as the total time during the flight of the projectile then the *T**f** *will be twice the *T**m** *(*where y = 0 and x = max*).

The equation of maximum height *Hm* can be determined by substituting *t = tm* in the expressions of *y* as given above.

The horizontal range *(x = R)* is the distance travelled by a projectile from its initial position *(x = y = 0)* to the position where it passes *y = 0* during its fall.

It is the distance traveled during the time of flight *Tf*. It can be determined by substituting *t = Tf* in expressions of *x *as given above.

This equation shows that for a given projectile motion where velocity is u, R will be maximum when *sin2θ* is maximum, i.e., when *θ = 45*. The maximum horizontal range will be

Also, read about uniform circular motion...

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