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Projectile Motion : Definition, Equations, & FAQs

Jul 18, 2022
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Projectile Motion

In Physics, there are numerous motion concepts one must be aware of. In the same order, there exists another motion named projectile motion or motion of a projectile. As its name suggests, projectile motion means throwing an object and noticing its motion upon arrival on the ground. Here, gravity is the only force acting when the object falls.

In this Physics blog, let us discuss the motion of a projectile and various other interrelated concepts. In the meantime, we will take a good look at the following things:

parallel
  • Projectile motion equations
  • Horizontal range
  • Time of flight formula
  • Maximum height of the projectile
  • The equation of trajectory
  • Projectile’s Parabolic Motion
  • How does Physics work in Basketball?
  • Conclusion
  • FAQs

Let’s dive deep into it.

What is a Projectile?

A projectile is nothing but an object that is thrown into space. Here there is only one force acting, which is gravity. So, we can say that the primary force that acts on the body is the gravitational force. Also, it does not necessarily mean no other forces are acting on it. But those forces will be minimal and negligible compared to the force of gravity.

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The path in which the object follows is known as the trajectory. An ideal example would be hitting the ball with a baseball or a cricket bat. Here, the ball follows a certain path to reach the ground surface.

Projectile Motion – Definition

While throwing an object indirectly near the surface of the earth, it may move along a curved path. When the motion is taking place, the acceleration will be constant. The object is directed towards the earth’s centre, assuming the particle travels close to the surface of the earth. The path followed by the object or a particle is known as a projectile, and its motion is termed projectile motion.

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In addition, projectile motion consists of two rectilinear motions, which are simultaneously independent. They are listed below:

  • Motion along the x-axis: Here, the velocity is uniform, which is responsible for the forward or horizontal motion of the object.
  • Motion along the y-axis: Here, the acceleration is uniform, which is responsible for the downward or vertical motion of the object.

To sum up, acceleration will act in both horizontal and vertical projectile motion of the object. Consider an object projected in space with a certain speed. Acceleration due to gravity is the only force that acts on the object when it is in flight mode. Gravity can be denoted as ‘g’. The acceleration’s direction is vertically downward. It indicates that the particle’s velocity remains constant in the horizontal direction.

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Projectile’s Parabolic Motion

Imagine a ball is being projected at a certain angle named with respect to the x-axis (horizontal direction). Its initial velocity ‘u’ is given below:

Parabolic Motion

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From the graph, we name,

O as the point of projection

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as the angle of projection

OB as the horizontal range (or simply range)

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Time of flight is the total time taken by the particle when it reaches from O to B.

In order to find varied parameters related to the motion of the projectile, differential equations of motions are used:

  • v = u – gt
  • s = ut – 12 gt2
  • v2 = u2 – 2gs

where,

u = initial velocity

v = final velocity

t = time

s = displacement

g = acceleration due to gravity

Total Time of Flight

In the vertical direction, displacement, s=0. Hence, provided below is the time of flight formula with the help of the equation of motion:

gt2 = 2  (uyt – sy) [here, uy = u sin and sy = 0]

Which becomes gt2 = 2t × uθ

Thus, the formula for time of flight can be written as follows:

Total Time of Flight (t) = 2usinθ/ g

Horizontal Range

The common formula for calculating a horizontal range is:

Horizontal range 

= Horizontal component of velocity × (Total flight time) 

which gives OA = ux / t

R = uθ × 2u × sinθ / g 

Hence, the horizontal range given by a projectile motion is given by R:

Horizontal range (R)= u22θ/ g

Projectile’s Maximum Height

Now that we have understood what a projectile is, it is time to learn about the projectile’s maximum height. An object’s maximum height can be measured along its trajectory from the highest vertical position. The projectile’s horizontal displacement is named the range of the projectile. Furthermore, the projectile’s range completely depends on the object’s initial velocity.

Consider,

v as the initial velocity,

H as the maximum height (in metres),

g = acceleration due to gravity,

= initial velocity angle from the horizontal plane (in deg or rad).

Therefore, the following is the formula for the maximum height of the projectile:

H = v20 sin2θ/ 2g

Trajectory Equation

The motion of the projectile is always parabolic. Following is the projectile motion’s equation for trajectory:

Equation of Trajectory = x tan

How does Physics work in Basketball?

Projectile motion can be called a motion in a plane or a 2D motion. In this motion, we can assume that gravity is the only motion acting on the body or the particle. However, we should know what its real-world applications are in order to understand the concept completely. Let us see how:

Take the game, basketball. What happens in this game? Two teams would be playing against each other, having five players per team. In order to score points for the team, the players have to jump and throw the ball into the baskets, which are placed on either side of the court.

When the ball is in motion, it forms a projectile. Therefore, we can call this projectile motion. Jumping and throwing give more acceleration to the ball than throwing while standing still.

Apart from basketball, take another example, such as throwing a cricket ball, a javelin throw, a stone in a lake, playing football, or an angry bird. The one thing that is common among all these activities is the projectile motion. It means that right after they were released, the only force acting on them would be the gravitational force, which pulls them into a downward position. Furthermore, it provides them with an equal and impartial acceleration.

Moreover, it indicates that when something is thrown in the open sky, we can easily predict how long it will stay up in the air and how much distance it will cover before reaching the ground. While throwing in a horizontal direction, there will be no further acceleration if we neglect the air resistance.

It implies that when a body is thrown near the ground, the motion acting on it is a 2D motion. It consists of acceleration in only one direction. So, we might wonder how this body follows a 2-dimensional path. In order to comprehend that, consider a rolling ball like in the figure:

motion in one dimension

The given figure indicates the motion of a ball in one dimension. However, since the graph has two coordinates, x and y, we may think that the ball follows two dimensions. But it is wrong. We can see a path in which the ball travels. Hence, it is a perfect example of a one-dimensional motion. From that, we should understand that the axes do not matter; only the motion’s nature does.

motion in plane

The given figure is an example of a motion in a plane. Consider throwing the ball at a certain angle. The ball’s velocity consists of x, y and z components. However, it does not mean that it follows a 3D motion. Only a plane can define the motion of an object, not a line. 

Hence, while throwing a body at any angle, there will be a plane that fully contains the body’s motion. From that, we can conclude that as long as the body is near the surface of the earth, it follows a two-dimensional motion neglecting the air resistance. It does not matter how many axes are chosen. 

Let us see what it would look like while turning the axes, as shown below:

body's motion

Therefore, the dimensions of the body’s motion are concluded by the minimum number of coordinates essential for determining the body’s motion.

Conclusion

All in all, this article has provided a clear view of the topic called the motion of a projectile while going through its relative concepts such as Projectile motion equations, Time of flight formula, Horizontal range, The equation of trajectory and Maximum height of projectile. 

Frequently Asked Questions 

1. Give some examples of projectile motion.

  • Some of the real-time examples of projectile motion are:
  • Flowing of water from a tap
  • Throwing a stone in the pond
  • Tossing a coin
  • Throwing a cricket ball
  • Throwing a basketball ball inside the basket
  • Jumping from a bus
  • Bullets firing from a gun
  • Rocket launch

2. Briefly talk about two types of projectiles.

In a projectile motion, there are two components present. They are vertical and horizontal motions. Vertical motion takes place up and down the surface of the earth, while horizontal motion acts forward and backward.

3. What are the types of projectile motion?

The following are the three types of projectile motion:

  • Horizontal projectile motion
  • Oblique projectile motion
  • Projectile motion on an inclined plane
Projectile motion

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