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In projectile motion, the path of an object under the influence of gravity can be described using equations of motion. These equations explain how the object's position changes over time.

• In the horizontal direction, the movement is at a constant velocity, so the equation is simply: \[ x = v_{0x} t \] This tells us that horizontal displacement (\(x\)) grows linearly with time (\(t\)).
• In the vertical direction, gravity affects the motion, causing the object to accelerate downward. The vertical motion equation is: \[ y = v_{0y} t - \frac{1}{2} g t^2 \] Here, \(y\) represents vertical displacement, \(v_{0y}\) is the initial vertical velocity, and \(g\) is the acceleration due to gravity.

Understanding these equations is crucial because they help us analyze the separate components of a projectile's journey. They enable us to predict where the projectile will be at any given time.

The initial velocity of a projectile sets the stage for its entire motion. It is usually given as a vector with two components: horizontal and vertical. In this scenario, the initial velocity is described by \( \mathbf{v}_0 = \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} \).

• The horizontal component, \(v_{0x}\), is 1 m/s. This indicates how fast the projectile moves along the ground.
• The vertical component, \(v_{0y}\), is 2 m/s. This determines how quickly the projectile rises or falls.

Grasping the concept of initial velocity is essential, as it defines both the speed and direction of a projectile's path from the outset. Identifying these components allows us to separate the motion into horizontal and vertical movements, which can be individually analyzed.

The trajectory equation describes the path that a projectile follows through space. By using the components of the initial velocity and equations of motion, we can derive this equation.In our example:- We know \( x = t \) from the horizontal motion equation.- Substituting \( t = x \) into the vertical equation gives \( y = 2x - 5x^2 \).This shows how the projectile's vertical position \(y\) changes as a function of its horizontal position \(x\). Understanding this trajectory equation is fundamental because it provides a complete picture of the projectile's path. Each term in the equation has a specific role:- The linear term, \(2x\), represents the initial vertical velocity's contribution.- The quadratic term, \(-5x^2\), captures the effect of gravity over time.This form helps predict where the projectile will be at any point along its flight, offering valuable insights for solving related physics problems.