Projectile Motion Formula

There are various formulas for the Projectile Motion for calculation of various this such as:

• Time of Flight
• Horizontal Range
• Maximum Height

Let’s discuss these formulas with various different cases as follows:

Time of Flight of Projectile Motion

Time of flight is the total time taken by the projectile from start to end. We can calculate it as,

In the Y direction total displacement (S_y) = 0

Taking motion in Y direction only, 

S_y  = u_yt – 1/2(gt²)   

For object to achive peak height u_y = u sinθ and S_y = 0, and t it the time taken by object to achive the peak height.

0 = usinθ – 1/2(gt²)

⇒ t = 2usinθ/g

Time of Flight (2t) = 2usinθ/g

Now there can be various cases of the above-mentioned formula, let’s consider the following cases:

Case 1: If θ = 90°  

As we can see from the formula of Time of flight, time taken by the projectile is directly proportional to the angle of projection. For any given initial velocity(u) will be constant and g is always constant i.e., g=-9.8 m/s².

When projectile is projected at an angle of  90° time of flight will be maximum.

t_max = 2usinθ/g = 2u/g 

 [As sin 90° = 1]

Case 2: If θ = 30°

When the projectile is projected at an angle of 30° time of flight is half of the t_max as sin30° = 1/2.

t = 2usin30°/g = t_max/2 

Horizontal Range of Projectile

The horizontal range is the distance covered by the projectile horizontally and it can be calculated by the distance = speed/time formula, where speed is the horizontal component of initial speed or velocity and time is the total time of flight. Thus, the formula for Horizontal Range is given by:

Range (R) = u_x × t

And as u_x = u cosθ and t = 2usinθ/g

Range (R) = ucosθ × 2usinθ/g

As a result, the Horizontal Range of the projectile is given by (R):

Horizontal Range (R) = u²sin2θ/g

Now there can be various cases of the above-mentioned formula, let’s consider the following cases:

Case 1: If θ = 90°  

When projectile is projected at an angle of 90° Horizontal range will be zero, because projectile will strike at the same point where the projectile is projected. 

R  = u²sin2θ/g = 0

[As sin 2θ = sin 180 = 0, at θ = 90°]

Case 2: If θ = 45°

When projectile is projected at 45° Horizontal Range of the projectile is maximum.

R_max = u²sin2θ/g = u²/g

As sin 90 = 1 and it is the maximum value of the trigonometric ratio sin.

Maximum Height of Projectile 

It is the highest point of the particle (point A). When the ball reaches point A, the vertical component of the velocity (V_y) will be zero. 

 0 = (usinθ)²  – 2gH_max  

 [ Here, S = H_max , v_y = 0 and u_y = u sin θ ]

Therefore, the Maximum Height of the projectile is given by (H_max):

Maximum Height (H_max) = u²sin²θ/2g

Now there can be various cases of the above-mentioned formula, let’s consider the following cases:

Case 1: if θ = 90°

If we project a projectile at an angle of 90° it achieves maximum height (H_max).

H_max = u²sin²θ/2g = u²/2g

[As, sin² 90° = 1 ]

Case 2: if θ = 45°

When the projectile is projected at an angle of 45°, the height of the projectile is half of its maximum height (Hmax) as sin²45° = 1/2.

H  = u²sin²θ/2g = (1/2)u²/2g = H_max/2

We can also say that if the projectile angle is 45° than Horizontal range of projectile will be 4 time the height of projectile.

H = u²/4g = R/4

OR

 R  = 4H

[ As Horizontal range at θ = 45°, R = u²/g ]

Projectile motion refers to the curved path an object follows when it is thrown or projected into the air and moves under the influence of gravity. In this motion, the object experiences two independent motions: horizontal motion (along the x-axis) and vertical motion (along the y-axis).

Projectile Motion can be seen in our daily life very easily as from throwing a rock to launching a cannonball are all examples of Projectile Motion. It is one of the fascinating topics in the field of physics which has very wide real-world applications. From sports to military technologies all leverage the understanding of Projectiles and their motion under the force of gravity.

Understanding Projectile motion helps us predict the trajectory, velocity, and range of objects that are thrown, launched, or dropped in the air. In this article, we will learn the key concepts and formulas of projectile motion and use those to solve real-world scenario-based problems.