Trajectory Formula: Projectile Motion, Trajectory Path, Derivation, Examples, Sample Questions

A trajectory is the flight path or course followed by an object that is shot in the air under the influence of gravity. This object is termed as a projectile and it undergoes a two dimensional motion by forming a parabolic curve. The real-life examples of trajectory in sports include football, soccer, hockey, and javelin throw, etc. The projectile is propelled with an initial velocity, which keeps it in motion, irrespective whether that projectile is thrown upwards or downwards, the only force acting on it is the gravitational force. The two components of motion here are independent of each other.

Key Terms: Trajectory, Path, Projectile, Parabolic, Gravity, Velocity, Motion, Horizontal, Vertical, Gravitation, and Force.

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Projectile

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A projectile is an object set in flight by applying an external force. The projectile may be thrown up vertically or at an angle to the horizontal. It may be dropped from a position of rest. The only force acting on the projectile during its motion along the flight path is the gravitational force and it is in motion due to its own inertia. 

Types of Projectiles

Examples of projectile are a ball hit by a bat, bullet fired from a gun, shell launched from a launcher, bomb dropped from a plane, etc. It must be noted that a rocket or a missile cannot be considered as a projectile as they are propelled by power.

Also Read: Displacement vector

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Trajectory path / projectile motion

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Motion Components along a trajectory path

The path of flight of a projectile is called its trajectory. It is a parabolic curve. In order to analyze the projectile motion or the trajectory, it is split into two components namely - vertical and horizontal. The projectile moves horizontally under the influence of the horizontal force acting on it and it moves downwards under the effect of the gravitational pull exerted on it. The projectile follows a parabolic curved path under the influence of a combination of these forces as seen in the image above.

It can be seen in the image that the projectile path will be a straight horizontal line if it is only acted upon by the horizontal component. Whereas, the path will be a vertical line if it is only under the influence of gravitational acceleration. Thus, the parabolic curved path is achieved only under the combined horizontal and vertical components of velocity.

Also Read: Uniform Circular Motion

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Trajectory Formula Derivation

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Trajectory formula derivation

When a ball is kicked from the ground, it starts its journey at an initial velocity and angle of launch with respect to the horizontal ground. The flight of the ball as gravity acts upon it follows a curved path or a parabola, this curve is called trajectory of the ball.

• The ball trajectory will have two dimensional components namely horizontal (x) and vertical (y) position. 
• Considering the initial velocity of Vo and launch angle θ, the vertical position component can be calculated based on the horizontal position by applying the following trajectory formula:

Vertical position = (horizontal position) (tangent of launch angle) – \(\frac{(\text{acceleration due to gravity})(\text{horizontal position})^2}{2(\text{initial velocity})^2 (\text{cosine of launch angle})^2}\)

y =xtanθ – \(\frac{gx^2}{2v^2cos^2\theta}\)

Where,

y is the vertical position component 

x is the horizontal position component

g= gravitational acceleration (9.80 m/s²)

v= initial velocity,

θ = angle of launch 

The Trajectory related equations are:

Where,

V_o is the initial Velocity,

Sinθ is the y-axis vertical component,

Cosθ is the x-axis horizontal component.

Also Read: Resultant Vector Formula

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Explanation of the projectile motion

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Projectile motion

As shown in the image above the path followed by the projectile is parabolic. Under the effect of the initial velocity the projectile reaches the maximum height it can attain and then under the influence of gravitational acceleration it begins its downward motion. 

• The falling projectile now gains a downward acceleration affecting the vertical component of velocity. 

• It must be noted that under negligible air resistance which accounts for zero horizontal forces, the horizontal component of velocity remains constant throughout.

• The horizontal motion of the projectile is due to its own inertia as it follows Newton’s first law of motion.

Also Read: Centripetal Acceleration

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List of equations for projectile motion

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Things to Remember 

• The path followed by a projectile is a parabolic curve.
• The trajectory motion of a projectile is two dimensional.
• The motion of projectiles is analyzed in terms of two independent motions perpendicular to each other.
• The range (R) of the projectile is the distance from the point of launch to the point on ground where it ends its journey. 
• The initial velocity is dependent upon the initial acceleration force with which the projectile is launched

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