Motion in a plane 4

Projectile:
A projectile is any object that once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

Or

A projectile is any object which after being thrown falls under the effect of gravitational force only.

Projectile Motion Path equation derivation | parabola

Let’s start with the derivation of Projectile Motion Path Equation. Say an object is thrown with uniform velocity V₀making an angle theta with the horizontal (X) axis.

The velocity component along X-axis = V_0x = V₀ cosθ   and  the velocity component along Y-axis = V_0y = V₀sinθ. Air resistance is taken as negligible.

At time T = 0, there is no displacement along X and Y axes. So X₀=0  and Y₀=0

At time T=t,

Displacement along X-axis = x= V_0x.t 
= (V₀ cosθ). t ………… (1) and

Displacement along Y-axis = y= V_0y.t 
= (V₀sinθ ).t  – (1/2) g t² …………(2)

From equation 1 we get: t = x/(V₀ cosθ) ………….. (3)

Replacing t in equation 2 with the expression of t from equation 3:

y = (V₀sinθ ). x/(V₀ cosθ)  – (1/2) g  [ x/(V₀ cosθ)]² 
or, y = (tanθ) x –   (1/2) g . x²/(V₀cosθ)²………..(4)

In the above equation g, θ and V₀ are constant.

So rewriting equation 4:
y = ax + bx² where a and b are constants.

This is an equation representing parabola.

So we can say that the motion path of a projectile is a parabola.

So once thrown a projectile will follow a curved path named Parabola.

Time of Flight:

Time taken by projectile to complete its trajectory is called as time of flight.

Time taken to reach the Maximum Height by a projectile

When the projectile reaches the maximum height then the velocity component along Y-axis i.e. V_y becomes 0.

Say the time required to reach this maximum height is t_max .

The initial velocity for the motion along Y-axis (as said above) is V₀sinθ

Considering vertical motion along y-axis:

V_y = V₀sinθ  – g t_max
=> 0 = V₀sinθ  – g t_max
=> t_max= (V₀sinθ )/g     ……………….(5)

So this is the equation for the time required to reach the maximum height by the projectile.


 Total Time of flight for a projectile:

So to reach the maximum height by the projectile the time taken is  (V₀sinθ )/g

It can be proved that the projectile takes equal time [ (V₀sinθ )/g] to come back to the ground from its maximum height.

Therefore the total time of flight for a projectile T_tot = 2(V₀sinθ )/g …………………. (6)

Maximum Height reached by a projectile

The maximum vertical distance attained by a projectile is called as Maximum Height.

Let’s say, the maximum height reached is H_max . We know that when the projectile reaches the maximum height then the velocity component along Y-axis i.e. V_y becomes 0.

Using one of the motion equations, we can write

(V_y)² =( V₀sinθ )² – 2 g H_max

=> 0 = ( V₀sinθ )² – 2 g H_max

=> H_max = ( V₀sinθ )²/(2 g) ……………… (7)

Horizontal range of a projectile

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

Say R is the horizontal range covered by a projectile is R when it touches the ground after passing the time Ttot

Here we will use the horizontal component of the velocity only as the effective velocity to traverse this horizontal path.

R = (V₀ cosθ) .  Ttot  
= (V₀ cosθ). 2(V₀sinθ )/g = (V₀² sin2θ )/ g

R = (V₀² sin2θ )/ g ………….. (8)

For the Maximum value of R, it is pretty easy to understand from the above equation that the angle θ needs to be equal 45 degrees, because in that case, we get the maximum value from sine as 2θ becomes 90 degrees.

Hence R_max = V₀² / g  ………………….. (9)


Here we have discussed the Projectile Motion Derivation steps to derive a set of equations or formula. We will add some numerical as well very soon.

List of Projectile motion formula or equations 

Motion Path equation:  y = (tanθ) x –   (1/2) g . x²/(V₀ cosθ)²

Time to reach max height:  t_max= (V₀sinθ )/g

Total time of flight for a projectile T_tot = 2(V₀sinθ )/g

Maximum height reached:  H_max = ( V₀sinθ )²/(2 g)

Horizontal range of a projectile:  R = (V₀² sin2θ )/ g

Maximum possible horizontal range:R_max = V₀² / g