Projectile Motion

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Projectile motion is an example of curved motion with constant acceleration. This is the two dimensional motion of a particle thrown obliquely into the air. The ideal motion of a cricket ball, a golf ball or a bullet is an example of projectile motion. We assume that the effect air could have on their motion is negligible. Moreover, we also neglect

• The effect due to curvature of the Earth

• The effect due to rotation of the Earth

For all points on the trajectory

The acceleration due to gravity 'g' is constant in magnitude and direction.

Types of projectiles

Horizontal projectile

If a body is projected horizontally from a certain height with a certain velocity, then the body is called a horizontal projectile.

Oblique projectile

If a body is projected at a certain angle with the horizontal, then the body is called an oblique projectile. The motion of a projectile is a two dimensional motion. So it can be discussed in two parts

• Horizontal motion

• Vertical motion

These two motions take place independent of each other. This is called the principle of physical independence of motion.

At any instant, the velocity of a projectile has two components

• Horizontal component

• Vertical component.

The horizontal component remains unchanged throughout the flight. The vertical component is continuously affected by the force of gravity. Therefore, while the horizontal motion is a uniform motion, the vertical motion is a uniformly accelerated motion.

Horizontal projectile

The figure above illustrates a body thrown horizontally from a point O with a velocity The point O is at a certain height above the ground. Let x and y be the horizontal and vertical distances covered by the projectile, respectively, in time t. Therefore, at time t, the projectile is at p.

In order to calculate x, let us consider the horizontal motion, which is uniform motion. This is because the only force acting on the projectile is the force of gravity. This force acts vertically downwards and hence, the horizontal component in zero. Therefore, the equations of motion of the projectile for the horizontal direction is just the equation of uniform motion in a straight line.

x = vt ------------------ (i)

In order to calculate y, the vertical motion of the projectile is considered. Since the vertical motion is controlled by the force of gravity, it is an accelerated motion. The initial velocity, v_y (0), in the vertically downward direction is zero. Since the Y-axis in the figure above is taken downwards, the downward direction is regarded as the positive direction. So, the acceleration of the projectile is + g.

From the equation

Here v_y (0) is taken as zero because both distance and time are being measured from the origin O.

From equation (1)

Substituting for t from the above equation in equation (2) we have,

is a constant for a projectile projected upwards with a definite velocity v and at a place with a definite value of 'g'.

Equation (3) is a second-degree equation in x, a first-degree equation in y and is the equation of a parabola.

Therefore, a body thrown horizontally from a certain height above the ground follows a parabolic trajectory till it hits the ground.

Resultant velocity of a horizontal projectile:

In this section, let us calculate the resultant velocity of the projectile , at any point p on the trajectory, in an interval of time t. V_x and V_y are the horizontal and vertical components of as illustrated in the figure below.

 

 

The magnitude of the resultant velocity is given by,