How Does Projectile Motion Work? Complete Physics Guide for Students
We all know from Newton’s Universal Laws of Gravity that any object on Earth is constantly influenced by gravity and pulled towards the center of the gravitational field, that is the center of the Earth. Gravity influences any object suspended in the air to fall downwards, as in the case of fruits from trees. Also, if you launch an object up into the air, it travels into the air due to its inertia of motion. But since the object is also under the influence of gravity, it also dips down in the ground. Due to the influence of gravity, that object takes a curve path forward towards the ground before it eventually stops after touching the ground. The object which is launched into the air is called projectile, and this kind of motion is called projectile motion.
When we throw an object up, due to the force of gravity it eventually comes down. An object which is in motion in the air with no other force but gravity influencing it is a projectile. We can define a projectile as an object which is projected into the air with only downward gravitational force influencing it and continues its motion by the virtue of its own inertia.
Projectiles can roughly be classified into three kinds:
A projectile which is left to free fall from a considerable height
A projectile which is launched directly upwards
A projectile upwards at an angle towards the horizontal
What Is Projectile Motion?
Projectile motion is the motion experienced by an object in the air only under the influence of gravity. A projectile, that is launched into the air near the surface of the Earth’s and moves along a curved path, or in other words a parabolic path, under the action of gravity, assuming the air resistance is negligible. Study of such motion is called ballistics. The only force acting upon a projectile is gravity, which imparts a downward acceleration to the projectile. Because of the object’s initial inertia, an external force is unnecessary to maintain the horizontal velocity of the object. It also requires considering other external forces like friction from aerodynamic drag or internal propulsion when the projectile motion is at a larger scale.
When you throw a particle in an oblique path near the surface of the earth, it follows a curved route with constant acceleration. In such a situation, the particle's path is known as a projectile, and its motion is known as the projectile motion.
In a projectile motion, two separate rectilinear motions occur at the same time:
A particle's horizontal and vertical projectile motions are accelerated in the following ways: When a particle is sent into the air at a certain speed, gravity's acceleration is the only force acting on it during that period (g). Along the x-axis is uniform velocity, which is responsible for the horizontal particle, also known as the forward motion.
Along the y-axis: uniform acceleration, which is responsible for the particle's vertical (downwards) motion. This downhill acceleration has a vertical component. There is no acceleration in the horizontal direction, implying that the particle's velocity in that direction remains constant.
Maximum Projectile Height
Let us know the maximum height of a projectile after grasping what a projectile is and how it moves. The highest vertical location along the object's flight determines its maximum height. The object's starting velocity determines the projectile's range.
If v is the beginning velocity, g is gravity's acceleration, and H is the most significant height in metres, then = the initial velocity's angle from the horizontal plane (radians or degrees).
The formula that has been derived for calculating the maximum height of a projectile is:
H=\[\frac{usin\theta }{g}\]
Ballistics, the study of projectile motion:
The science of mechanics that deals with the flight, behaviour, and effects of projectiles are called ballistics. The word ballistics comes from the word ballista, which was a siege machine used for war during the late 18th century. Ballista was a weapon that is used to hurl boulders or spears against the enemy troops or walls. The boulder or spears used against the enemy were launched by the ballista, after which it would travel in a parabolic path and land among the enemy. The operators of the ballista have to be concerned about the angle to launch the projectile and the path they had to travel to gauge the landing position of the ballista. This is the reason the study of projectile motion is called ballistics. The path travelled by the projectile after it is launched is called the ballistic trajectory. The study of ballistics is important in modern warfare since it involves calculating the trajectories of modern warfare equipment like bullets, unguided bombs, rockets, and such other equipment. It also involves the designing of projectiles to attain desired acceleration and performance.
The elementary equations of ballistics only take into account the initial velocity of the projectile and an assumed constant gravitational acceleration. Practical applications of ballistics have to consider a variety of other factors like air resistance, crosswinds, target motion, the varying acceleration due to gravity, and in the case the target is significantly far away, even the rotation of the earth.
The weapon with the highest potential for destruction is named the ballistic missile since it involves delivering warheads to a target by a missile which follows a ballistic trajectory and falls on the target under the influence of gravity.
Some Important One-Liners to Remember About Projectile Motion
The kinetic energy is , and the linear momentum is m.u.cosፀ at the highest point.
The horizontal displacement of the projectile after t seconds is
The vertical displacement of the projectile after t seconds is
is the equation for the projectile's path.
A projectile's course is parabolic.
The kinetic energy is at the lowest position.
The linear momentum is equal to m.u at the lowest position.
The projectile's acceleration is constant during the motion and operates vertically downwards, equal to g.
The angular momentum is equal to where h is the height.
In the case of angular projection, the angle between velocity and acceleration might range from 0 to 180 degrees.
Key components which are to be remembered while dealing with projectile motion:
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In the given diagram
The key components which we should remember when calculating projectile motion and solving problems are
Initial launch angle, or angle of projection
The initial angle of projection varies from 0° to 90° And it heavily influences the motion of a projectile.
Initial velocity, u
If the path of the projectile, its launching point, and its target is plotted on a graph, initial velocity can be expressed as x and y components. The initial velocity of the projectile, in that case, can be given as
uxor uy = u sinፀ
or we can use the Pythagoras theorem to find the velocity
Time of flight, T
The time of flight is the time period from when the object is launched to the time it reaches the ground. The time period the projectile is in the air depends on the initial velocity and the angle of projection. It is given as
T = \[\frac{2usin\theta }{g}\]
Acceleration, a
There is no acceleration in the horizontal direction in case of projectile motion. The only acceleration present is the downward acceleration, that is acceleration due to gravity, also known as free fall. It is given as
ax = 0 or ay = -g
The value of g is equal to 9.8 m/s
Horizontal velocity, ux, and Vertical velocity, uy
At any point in the projectile motion, the horizontal velocity remains constant. On the other hand, vertical velocity varies linearly. This is because acceleration is constant at 9.8 m/s
Displacement, d or position at time t
At time given by t, the displacement components in a graph plotted with the origin of the projectile as the origin, the displacement components are
X = u.t.cosፀ and y = u.t.sinፀ-gt²
Maximum height, H
The maximum height of the projectile is the highest height the projectile can reach
It is given by
H = \[\frac{u^2sin^2\theta }{2g}\]
Range, R
The range of a projectile is the distance between the launch point and the target in a straight line.
It is calculated by R = \[\frac{u^2sin2\theta }{g}\]
FAQs on Projectile Motion Made Simple: Definitions, Formulas & Examples
1. What is projectile motion in simple terms?
Projectile motion describes the path of an object that is thrown, or projected, into the air and then moves only under the influence of gravity. After its initial launch, no other force acts on it, assuming we ignore air resistance. The path it follows is called its trajectory.
2. What are the two main components of projectile motion?
Projectile motion is analysed by splitting it into two independent components:
- Horizontal Component: The velocity in this direction remains constant throughout the motion because there is no horizontal acceleration (assuming no air resistance).
- Vertical Component: The velocity in this direction changes continuously due to the constant downward acceleration of gravity (g). It decreases as the object rises and increases as it falls.
3. What is the shape of a projectile's path and why?
The path or trajectory of a projectile is a parabola. This characteristic shape results from the combination of its two motion components: the constant-speed horizontal motion and the uniformly accelerated vertical motion. The superposition of these two motions creates the curved parabolic path.
4. What are the key formulas used to describe projectile motion?
The three primary formulas for a projectile launched at an angle θ with initial velocity u are:
- Maximum Height (H): H = (u² sin²θ) / 2g
- Time of Flight (T): T = (2u sinθ) / g
- Horizontal Range (R): R = (u² sin2θ) / g
5. How does the projection angle affect a projectile's range and maximum height?
The angle of projection is a critical factor:
- Maximum Height: The maximum height increases as the launch angle increases, reaching its highest point at a 90° (vertical) launch.
- Horizontal Range: The range is maximized at a launch angle of 45°. Furthermore, for any two complementary angles of projection (e.g., 30° and 60°), the horizontal range will be the same, assuming the same initial velocity.
6. What is the velocity of a projectile at its maximum height?
At its highest point, the projectile's vertical velocity component is momentarily zero as it stops moving upwards and begins to fall. However, its horizontal velocity component remains constant and unchanged from the start. Therefore, the projectile's overall velocity at the maximum height is not zero; it is equal to its initial horizontal velocity (u cosθ).
7. Why is a car turning on a flat, curved road not an example of projectile motion?
A car turning on a flat road is not in projectile motion because its movement is restricted to a horizontal plane. Projectile motion requires an object to be moving through the air under the influence of gravity, which means it must have a vertical component of motion. The car's motion is governed by friction and centripetal force, not by free-fall due to gravity.
8. What key assumptions are made when studying projectile motion in school-level physics?
To simplify the calculations, we make several key assumptions:
- Air resistance is negligible and has no effect on the motion.
- The acceleration due to gravity (g) is constant in both magnitude and direction at all points along the trajectory.
- The rotation and curvature of the Earth are ignored.
9. What are some common real-world examples of projectile motion?
You can observe projectile motion in many everyday scenarios, including:
- A javelin or shot put thrown by an athlete.
- A basketball being shot towards a hoop.
- A cannonball fired from a cannon.
- Water jetting out from a fountain at an angle.
