Introduction to Projectile Motion
A projectile motion is the motion of any object that is thrown to the air or released in air. In kinematics, we study the various types of motion, like linear motion and projectile motion. The linear motion covers one-dimensional information about the motion whereas the projectile gives us the information about a two-dimensional motion. The motion of the projectile will totally be under gravity. Hence, an object is regarded as a projectile, if it is launched in air and moves only due to gravity. For example, a ball thrown upward in air is a projectile as after a certain time it starts to fall under gravity (ignoring air resistance). The path followed by the projectile is known as trajectory, which is usually curved.
Several applications like from a simple football and cricket game to the bomb explosion and sharp shooting, are based on projectile motion. Hence, its study is fundamental to understand the two-dimensional motion of the objects that are under the free-fall.
Characteristics of a Projectile
The major characteristics that differ projectile from the motion of other objects are given below:
- Once launched, its initial final velocity must be zero and should fall only under gravity. Here, the velocity of the projectile is resolved into two components i.e horizontal component of velocity along x-axis, which is not affected by gravity and, the vertical component which is solely the effect of gravity.
- The path followed by the projectile must be curved, as it shows the combined effect of horizontal and vertical motion.
- The motion must be two dimensional i.e. it covers a certain horizontal range along the x-axis and a vertical height along the y-axis at the same time.
- The object finally fell vertically downward and hit a target.
Types of Projectile Motion
Projectile motions are of two types, which are given below:
- Horizontal Projectile Motion
This is a projectile which is dropped straight by making a 0° angle with the horizontal, from a certain height. For example: a bag just dropped from the roof or a bullet fired from the cliff with 0° with the horizontal. Here, initial velocity will be always zero and the horizontal velocity is always constant. The vertical velocity is affected by the gravity and increases gradually. The path followed by the projectile is parabolic.
- Vertical Projectile Motion
This is a projectile, launched vertically upwards or downwards with an angle 90° with the horizontal. Example: Throwing a ball at an angle from the ground. Here too, the horizontal velocity is constant, but the vertical velocity keeps changing due to gravity. The path is also parabolic. For example, a football kicked from the ground.
- Oblique Projectile Motion
In this type of projectile motion, the projectile is launched at other angles than 0° or 90°. The horizontal motion is not affected by gravity and vertical motion keeps changing. The path is the same as above i.e. parabolic. For example, a suitcase dropped from an airplane, making an angle 60° with the horizontal.
- Projectile Motion on an Inclined Plane
This projectile motion is quite more complex than the former once. It deals with the projectile which is launched from a sloped surface. The motion is directly affected by the projection angle as well as the inclination angle. The velocity and acceleration due to gravity, both are resolved into vertical and horizontal components for the calculation. For example, the motion of a rock, falling downhill.
For each case, the equations of motion are different.
Some Assumptions in Projectile Motion
There are some simple assumptions, made for simpler calculations which are given below:
- Air resistance is ignored which means that the projectile is not affected by air.
- Acceleration due to gravity (g) is always taken constant i.e. 9.8 m/s² and is assumed to be acting downward.
- Horizontal velocity component is constant and hence horizontal acceleration is zero. Thus, no force acts horizontally.
- The point of projectile launch is taken as the origin.
Velocity Components and Equations of Motion
As described above, a projectile has motion along both the vertical and the horizontal. Both motions are different in nature and hence, the calculation of motion will be easier if the vertical and horizontal motions are treated differently. Since, gravity only occurs for falling objects (i.e. downward or upward motion) the horizontal motion remains untouched by the gravity. Hence, at every point of projection of a projectile the velocity is resolved itno two components i.e. horizontal component and vertical component. So, if a projectile is launched, at initial condition the initial velocity can be regarded as ‘u’ which makes angle θ with the horizontal. Therefore, the velocity is resolved as:
Horizontal component
ux = ucosθ
But along horizontal θ = 0 and u = ux
As the horizontal range goes straight along the x-axis, the horizontal component remains constant throughout the motion.
Vertical component
uy = usinθ
As the vertical height keeps changing due to gravity, the vertical component also varies at each point of the trajectory.
Some equations used for calculations along each direction are given below:
Horizontal motion
We know, the equation for displacement is given by:
s = ut + ½ at^2 [Equation 1]
Where, u is the initial velocity, a is the acceleration and t is the time taken. For a projectile displacement along x-axis, since a = g is 0, and with the horizontal component ux = ucosθ, we get,
x = ucosθt [Equation 2]
Or t = x / ucosθ [Equation 3]
Vertical motion
As, the motion along the vertical is against the gravity (if we consider the body is thrown upward and), we have a = -g and vertical component of the velocity, uy = usinθ
y = usinθt−1/2gt^2 [Equation 4]
Substituting the value ot t from equation (3) in equation (4), we get:
y = tanθx−½ x^2 gu^2cos^2θ [Equation 5]
This is the equation of a parabola with a = tanθ and b = -g/2u^2cos^2θ. Hence, it shows that the path of a projectile is always parabolic in nature.Thus the two components of velocity combine to give a parabolic curve for the path. The projectile after thrown, starts to rise to a certain height, until the velocity becomes zero and then starts falling freely. However, the horizontal path is same for the rise and fall.
Time of Flight, Maximum Height, and Range
There are other major terms used in the calculation of projectile motion which are:
- Time of Flight (T)
It is the time fow which the projectile remains in the air.
Since, the projectile rise up and falls the same distance, the displacement will be zero.
Or s = usinθT−1/2gT^2
Or 0 = usinθT−1/2gT^2
Or T=2usinθ/g [Equation 6]
- Maximum Height (H)
It is the highest point reached by6 the projectile above the ground. The final vertical velocity at this point will be zero. Hence, vy = 0.
Or vy^2 = uy^2 – 2gH
Or 0 = u^2sin^2θ – 2gHmax
Or Hmax = u^2sin^2θ / 2g [Equation 7]
- Range (R)
It is the horizontal distance covered by the projectile, before hitting the ground. Since, there is no acceleration due to gravity here, thus,
R = ucosθT = ucosθ (2usinθ/g)
Or R = u^2sin2θ / g [Equation 8]
For maximum range, R = u^2 / g, θ = 45°.
We can conclude that the value of θ i.e. the angle of projection can greatly affect all these quantities: Shape of the trajectory, time of flight, maximum height and the horizontal range.
Note: In real life, air resistance cannot be ignored. Air can slow down the motion of projectile and hence the range and the maximum height can be reduced. Therefor the curve of the path can get asymmetric.
Real-Life Examples and Applications of Projectile Motion
Some real examples and practical applications of projectile motion are as follows:
- Sports
We can see the maximum use of projectile motion in sports. Games like football, basketball, cricket, volleyball, Javelin throw all show the projectile motion and the opponent catch can also be pre-calculated using the above formula tricks.
- Military
Another good application of projectile motion is in military. The missile attacks, firing and artillery shells use a wonderful implication of the physics of projectile motion.
- Engineering and Space Applications
In engineering, the rocket stage separation is made by the using the physics of a projectile. The concept of escape velocity is also equally applied before launching the rocket. Also, dropping supplies from aircraft is also done with the projectile calculations.
- Daily Life
Some other daily life facts implying projectile motion are, throwing a stone, spraying water, jumping off a swing etc. The mechanics of fountain is also based on a projectile.
Hence, understanding projectile motion can be used to build reliable safer and more productive technologies in all these aspects.
Conclusion
Projectile motion is like a basic aspect of daily life. Its mechanism has remained important not only in numerical calculations but a problem solution in the field of sports and military. Also, it has equal importance in science and engineering. The horizontal and vertical components combinedly give the symmetric path followed by a projectile. However, in the calculations air resistance has been neglected. This can effect greatly in the motion of the projectile where air resistance is greater. Otherwise, in normal places the simple calculations can be easily done in minutes if we know the projection angle and the velocities at different instants of time.
References
Projectile Motion
https://en.wikipedia.org/wiki/Projectile_motion
https://byjus.com/physics/projectile-motion
https://www.geeksforgeeks.org/physics/projectile-motion
https://www.physicsclassroom.com/Class/vectors/U3L2a.cfm
