A Short Study on Projectile Motion

Any object which is thrown into space and the only force acting upon it is gravity is called to be projectile, and the motion which the body undergoes is called projectile motion. The path undertaken by the projectile is known as trajectory.  

We tend to assume that the object is near the surface of the earth. When we say that only gravity acts on it, that doesn’t mean the other forces don’t. It means the effect of the other forces is minimal compared to gravity. 

Projectile motion is considered to be one of the most common motions in the plane. Any falling object can be considered an example of projectile motion.

Components of Projectile Motion

In projectile motion, there are two simultaneous rectilinear motions acting which are independent.

On the x-axis, there is a uniform velocity which is responsible for the horizontal motion of the body of the particle, which is a forward motion, and on the y-axis, there is uniform acceleration which is responsible for the vertical motion of the body of the particle. This is a downward motion.

Time of Flight

As the name itself indicates, the time of flight of a projectile motion is defined as the time from which the object is projected until the time it reaches the surface. It is dependent on the initial velocity of the object and the angle through which it is projected 𝛉. It can also be defined as the total time in which the projectile tends to remain in the air.

The projectiles tend to have parabolic motion, and for finding their parameters, we make use of the three equations of motions, i.e.

1. v = u – gt
2. s = ut + 1/2gt2
3. v2 = u2 – 2gs

In these equations, u is the initial velocity and v is the final velocity.

                                   g is the acceleration due to gravity

                                   t is the time

                                   s is the displacement                         

Types of Projectile Motion

There are basically three types of projectile motion, namely:

• Projectile motion that takes place on an inclined plane
• Horizontal projectile motion
• Oblique projectile motion

Total Time of Flight

In the vertical direction, the resultant displacement which is indicated by s is equal to zero and because of that, the formula to calculate the time of flight is given by using the motion equation.

gt2 = 2(uyt – sy), (In this equation, uy = usin𝛉 and sy = 0)

gt2 = 2t * usin𝛉

Therefore, the time of flight formula t is given by the equation

Total time of flight [t] = 2usin𝛉 / g

The Horizontal Range

The horizontal range is given as the product of the horizontal component of velocity and the total time of flight.

Horizontal range = Horizontal velocity component * Total flight time

R = ucos𝛉*2usin𝛉/g

Maximum Height of Projectile

The maximum height of the projectile is given when the object reaches its maximum height in the highest vertical position of its trajectory. The displacement which takes place horizontally is called the projectile range, and it is dependent on the initial velocity of the object. 

The formula for the maximum height of the projectile is

H = v2osin2𝛉 /  2g

Here, v is the initial velocity,

g is the acceleration due to gravity,

𝛉 is the initial velocity angle in the horizontal plane, which can be expressed in radians or degrees.

The Trajectory Equation

The trajectory equation of a projectile is given as

 y = x tan𝛉 – gx22u22

Relationship Between the Maximum Height and the Horizontal Range

The relation between the range d and the maximum height h, when reached at td/2, is given by, 

h = dtan𝛉/4

Angle of Reach

The angle at which the projectile must be launched to travel to a distance d, when the initial velocity v is given, is called the angle of reach

                                                          sin(2𝛉) = gd/v2

Here, 𝛉 is the angle of reach,  g is the acceleration due to gravity

Applications of Projectile Motion

In our day-to-day life, the uses of projectile motion are numerous. Some of them  are as follows:

• Sports like football and basketball are examples of projectile motion.
• Tossing some objects in the air or into the bin.
• Jumping from a bus while it slows down.
• The water fountains we see in different parks.
• The motion of all the heavenly bodies around the sun.
• Javelin’s throw and archery clearly show the projectile path.

Conclusion

Under constant acceleration, when an object is thrown into space, it moves along a curved path and is directed towards the centre of the earth. This kind of motion is called projectile motion. The only force acting upon the projectile motion is gravity; all the other forces are minimal, so they are not considered. 

Projectile motion has its own path, and that is called a trajectory. We see the use of projectile motion in our everyday lives, including in sports, water fountains, fireworks, etc. The trajectory equation acts as proof that all the projectile motions are parabolic in nature. We make use of the motion equations to find different parameters related to projectile motion.