The projectiles motion in 2 dimensions | POINT classical mechanics

 

projectiles motion in 2 dimensions

When a body (ball) is thrown upwards with an initial velocity (Vi) at an angle of inclination (θ) with the horizontal plane, it
takes a curved path under the influence of gravity as shown in the figure:

The velocity can be analyzed in the horizontal (x) and vertical (y) directions as follows:

In the horizontal direction (x)

Initial velocity: Vix = Vi cos θ

Final velocity using equations of motion: The ball moves at a constant velocity assuming no frictional forces

ax = 0

Vfx = Vix

In the vertical direction (y)

Viy = Vi sin θ

The ball moves at a constant velocity (Vy) under the influence of Free fall acceleration: ay = -

 Vfy can be calculated at any moment or at any height.

The ball's velocity is calculated at any moment from the Pythagorean theorem:

Deduce the ascent time t and flight time T: 

Vfy = Viy + gt

When the body reaches the maximum height, the velocity in the vertical direction (y) is zero, so we substitute (Vfy) in the first equation of motion, so it is: 0 = Viy + gt

The flight time T is the time between the start of the body's movement until it reaches the same level of its launch, twice the ascent time t: 

Deduce the maximum vertical height h:

When the body reaches the maximum height, the velocity in the vertical direction is zero (V = 0), but it has a velocity in the horizontal direction (V), From the third equation of motion: 2 ad = V²fy - V²iy   ,   2 gh = V²fy = - V²iy

Deduction of the maximum horizontal range R (the maximum horizontal distance traveled by the body):

The time the body reaches the maximum horizontal range = the flight time (T) // and by substituting (ax = 0), (d = R) in the second equation of motion:

vertically               horizontally

*The opposite figure shows the change in position of two balls over equal periods of time, where one of them was thrown horizontally while the other was dropped vertically at the same time, Neglecting air resistance. 

*For a ball that falls vertically freely in a straight line, it falls under the influence of its weight and its motion can be analyzed using the equations of motion with uniform acceleration in one direction Where a = g 

*As for the ball thrown horizontally, it moves with a constant horizontal velocity, a horizontal distance given by the relationship (Ax = vt), and its movement on the vertical axis is exactly like the movement of a ball falling Freely, since its initial vertical velocity is zero, it covers at any moment the same vertical distance covered by the ball falling freely, and for this reason we find that the two balls reach the ground at the same moment, and confirms the absence of a relationship between the distance of the fall and the horizontal component of the movement. 

The maximum horizontal range that a projectile reaches when thrown is at an angle of 45°.

*The horizontal range of a body projected at two different angles (θ1,θ2) and with the same initial velocity is equal when the sum of the two angles is equal to 90°.

*The summary of the above is: The movement of the projectile is a movement composed of a uniform velocity movement on the horizontal axis and a uniform acceleration movement on the vertical axis.

In summary, the study of free fall and projectile motion reveals the elegant simplicity underlying the laws of motion. By examining how objects accelerate under gravity alone or with an initial launch, we gain valuable insights into the fundamental principles governing motion. These concepts are essential not only for theoretical understanding but also for practical applications—from designing safer vehicles and sports equipment to predicting the paths of celestial bodies. Ultimately, the exploration of free fall and projectile motion deepens our appreciation for the natural order and consistency of the physical world.