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plot-trajectory-of-a-projectile

🚀 Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. In the context of motion in a plane, we specifically look at the trajectory of projectiles, which are objects that are thrown into the air and are subject to gravitational force. The path that a projectile follows is called its trajectory, and it is typically a parabolic shape due to the influence of gravity. Understanding the trajectory of a projectile involves analyzing its horizontal and vertical motions separately, as they are independent of each other.

Theory Explanation

Understanding Projectile Motion

Projectile motion can be analyzed by breaking it down into two components: horizontal motion and vertical motion. The horizontal motion is uniform, meaning the projectile moves at a constant velocity, while the vertical motion is uniformly accelerated due to gravity. The equations of motion for both components can be used to determine the trajectory of the projectile.

\[ x = v_0 imes t \]
Equations of Motion

For the vertical motion, we use the equations of motion under constant acceleration. The vertical displacement can be calculated using the formula: \( y = v_{0y} t - \frac{1}{2} g t^2 \), where \( v_{0y} \) is the initial vertical velocity and \( g \) is the acceleration due to gravity (approximately 9.81 m/s²). The time of flight can be calculated by determining when the projectile returns to the same vertical level from which it was launched.

\[ y = v_{0y} t - \frac{1}{2} g t^2 \]
Finding the Range

The range of the projectile is the horizontal distance it travels before landing. It can be calculated using the formula: \( R = \frac{v_0^2 \sin(2\theta)}{g} \), where \( \theta \) is the angle of projection and \( v_0 \) is the initial velocity. This formula derives from the horizontal and vertical motion equations and gives the maximum distance traveled by the projectile.

\[ R = \frac{v_0^2 \sin(2\theta)}{g} \]

Key Points

  • 🎯 Projectile motion is a combination of horizontal and vertical motions.
  • 🎯 The trajectory of a projectile is parabolic due to the influence of gravity.
  • 🎯 The horizontal motion is uniform, while the vertical motion is uniformly accelerated.

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Examples:💡

A projectile is launched with an initial velocity of 20 m/s at an angle of 30 degrees. Calculate the range of the projectile.

Solution:

Step 1: First, calculate the initial horizontal and vertical components of the velocity: \( v_{0x} = v_0 \cos(\theta) \) and \( v_{0y} = v_0 \sin(\theta) \).

\[ v_{0x} = 20 \cos(30^\circ), v_{0y} = 20 \sin(30^\circ) \]

Step 2: Calculate the range using the formula: \( R = \frac{v_0^2 \sin(2\theta)}{g} \).

\[ R = \frac{20^2 \sin(60^\circ)}{9.81} \]

A ball is thrown vertically upward with an initial velocity of 15 m/s. Calculate the maximum height reached by the ball.

Solution:

Step 1: Use the formula for maximum height: \( H = \frac{v_{0y}^2}{2g} \). Here, \( v_{0y} = 15 \) m/s.

\[ H = \frac{15^2}{2 \times 9.81} \]

Step 2: Calculate the maximum height: \( H = \frac{225}{19.62} \approx 11.47 \) m.

Common Mistakes

  • Mistake: Confusing the horizontal and vertical components of motion.

    Correction: Always separate the horizontal and vertical motions and apply the correct equations for each.

  • Mistake: Neglecting air resistance in calculations.

    Correction: Assume ideal conditions unless specified otherwise, but be aware that air resistance can affect real-world projectile motion.

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