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  Motion under Gravity, Physics tutorial

Concept of Projectile:

Whenever you release a piece of stone from your catapult against a bird perching on the tree branch, the stone travels in a parabolic path in the direction of the bird.

As well, if you throw a tennis ball against a wall, the path of the ball in the direction of the wall is a parabola. You will observe the similar kind of curve whenever you project a ball horizontally from the top of a building. The stone or ball you projected is termed as projectile, whereas the motion of the stone or ball projected is termed as projectile motion. The path followed through a projectile is termed as its trajectory.

Illustrations of projectile motion are the motions of:

  • Bullets fired from a gun or rifle
  • Thrown or kicked balls
  • Jumping animals
  • Bombs released from the jet
  • Objects dropped from the windows
  • An athlete accomplishing the high jump
  • Stone released from the catapult

Motion of Projectile:

Projectile is stated as, anybody thrown by certain initial velocity that is then allowed to move beneath the action of gravity alone, devoid of being propelled through any engine or fuel. The path followed through a projectile is termed as its trajectory. The projectile moves at a constant speed in the horizontal direction while undergoing a constant acceleration of 9.8 m/s2 downwards in the vertical direction. To be reliable, it is stated that the up or upwards direction to be the positive direction. Thus the acceleration of gravity is, -9.8 m/s2.

Vertical and Horizontal Projections:

Vertical Projectile Motion:

The velocity of body falling from a height 'h' on reaching the ground is equivalent to the velocity by which it is projected vertically upwards to reach the similar height 'h'. Therefore the upward velocity at any point in its flight is similar as its downward velocity at that point. We frequently toss or throw things directly upward and this is an illustration of the vertical projection. The initial velocity of object is up or upward however the acceleration due to gravity is downward. Therefore a vertically projected object at its maximum height stops instantly and changes or modifies its direction. Now it becomes a dropped object in the free fall. When a body is projected vertically upwards its velocity steadily reduces and if the body reaches the maximum height its velocity becomes zero. The equations of motion for an object that is thrown up or upwards are as shown below:

v = u - gt

s = ut - 1/2 (gt2)

v2 = u2 - 2gs

Here:

v = final velocity

u = initial velocity

g = gravitation acceleration

s = displacement

=> Points to remember:  

If an object is thrown vertically upward, then the maximum height attained through it is proportional to square of its initial velocity If an object is thrown vertically upward, time taken by it in reaching at the top point is equivalent to the time taken through it to reach back to the ground.

Horizontal Projectile Motion:

A horizontally fired bullet from a rifle and ball thrown held horizontally are the illustrations of projectiles in the horizontal direction. A body 'A' that is freely falling and the body 'B' projected horizontally from the similar height at similar time will hit the ground concurrently at dissimilar points. The two bodies will be at similar vertical point at any point of time.

For this kind of projection there is an initial velocity 'u' simply in the horizontal or x-direction.

However there is no initial velocity in the vertical or y-direction. Though, there is acceleration in the downward direction due to gravity. As there is no acceleration or force in the x-direction subsequent to it is projected, the projectile moves in this direction having a constant speed (u). As the object horizontally moves, it as well falls in the downward direction due to gravity. In the downward direction, the motion is similar as that of the dropped object.

Resultant Velocity of a Projectile:

We are familiar that the velocity of the projectile throughout the flight is made up of a horizontal component, 'u' and a vertical component 'ω'.

The vertical component 'ω' at time 't' after the beginning of the motion is provided by:

ω = gt

The resultant velocity:

v = √ (u2 + ω2)

v = √ (u2 + g2t2)

The angle Φ, between the direction of motion of the body and the horizontal is provided by;

tan Φ = ω/u = gt/u

Φ = tan-1 (gt/u)

Projection at an angle to the Horizontal:

The Projectile motion is a kind of motion where an object moves in a bilaterally symmetrical, parabolic path. The path which the object follows is termed as its trajectory. Projectile motion only takes place if there is one force applied at the starting on the trajectory, after which the mere interference is from gravity. The fundamental equations which go all along with them in the special case in which the projectile initial positions are null (that is, x0 = 0 and y0 = 0).

Initial Velocity:

The initial velocity can be deduced as x components and y components:

ux = u ⋅ cosθ

uy = u ⋅ sinθ

In the above equations, u signifies for initial velocity magnitude and θ signifies to the projectile angle.

Time of Flight:

The time of flight in a projectile motion is the time from when the object is projected to the time it arrives at the surface. As we are familiar that, T mainly depends on the initial velocity magnitude and the angle of the projectile:

T = 2⋅uy/g

T = (2⋅u⋅sinθ)/g

Acceleration:

In the projectile motion, there is no acceleration in the horizontal direction. The acceleration 'a', in the vertical direction is merely because of the gravity, as well termed as free fall:

ax = 0

ay = - g

Velocity:

The horizontal velocity stay constant, however the vertical velocity differs linearly, as the acceleration is constant. At any time 't' the velocity is:

ux = u⋅cosθ

uy = u⋅sinθ - g⋅t

We can as well utilize the Pythagorean Theorem to determine the velocity:

u = √ (u2x + u2y)

Displacement:

At time 't' the displacement components are:

x = u⋅t⋅cosθ

y = u⋅t⋅sinθ - (1/2) gt2

The equation for magnitude of the displacement is Δr = √ (x2+y2)

Parabolic Trajectory:

The equation for parabola is y = ax + bx2. Here, the displacement equations in x and y direction can be used to get an equation for the parabolic form of a projectile motion:

y = tanθ ⋅ x - [g/(2⋅u2⋅cos2θ] ⋅x2

Maximum Height:

The maximum height is obtained if vy = 0. By utilizing this we can rearrange the velocity equation to determine the time it will take for the object to reach the maximum height.

th = (u∗sinθ)/g

Here th signifies for the time it takes to reach the utmost height. From the displacement equation we can find out the maximum height.

h = (u2∗sin2θ)/(2∗g)

Range:

The range of motion is fixed through the condition y = 0. By utilizing this we can rearrange the parabolic motion equation to determine the range of the motion:

R = (u2∗sin2θ)/g

Applications of Projectiles:

From the knowledge and study of projectiles we can now figure out that the principle of projectiles is employed in warfare for aiming accurately at the targets, achieving maximum height and achieving maximum ranges. For illustration: we are familiar that for a bomber from an airplane to hit a target, the bomb should be released if the target appears at a certain angle of depression ~ given by:

tan Φ = 1/u √(gh/2)

It is as well familiar that at certain angle of projection (0 = 45°) a projectile would acquire a maximum range.

In sports, the knowledge of projectile motions could be helpful as maximum ranges can be attained in sports in short put, javelin, football, golf, baseball and so on. As well, obtaining maximum height, maximum time of flight is at times needed.

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