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derive-range-of-projectile-formula

๐Ÿš€ 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 projectile motion, we analyze the motion of an object that is thrown into the air and moves under the influence of gravity. The range of a projectile is the horizontal distance it travels before landing. This concept is crucial in physics and engineering, as it helps us understand how objects behave when launched at an angle.

Theory Explanation

Understanding Projectile Motion

Projectile motion can be broken down into two components: horizontal and vertical motion. The horizontal motion is uniform, meaning the horizontal velocity remains constant, while the vertical motion is influenced by gravity, resulting in a uniformly accelerated motion.

Deriving the Range Formula

To derive the range formula, we start by analyzing the time of flight and the horizontal distance traveled. The time of flight (T) for a projectile launched at an angle ฮธ with an initial velocity u is given by T = (2u sin ฮธ) / g, where g is the acceleration due to gravity. The horizontal range (R) can be calculated using the formula R = u * cos(ฮธ) * T. Substituting the expression for T, we get R = (u^2 * sin(2ฮธ)) / g.

\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Understanding the Variables

In the range formula, u is the initial velocity, ฮธ is the launch angle, and g is the acceleration due to gravity (approximately 9.81 m/sยฒ). The angle ฮธ plays a crucial role in determining the range, as it affects both the vertical and horizontal components of the initial velocity.

Key Points

  • ๐ŸŽฏ Projectile motion consists of horizontal and vertical components.
  • ๐ŸŽฏ The range is maximized at a launch angle of 45 degrees.
  • ๐ŸŽฏ The time of flight and range are dependent on the initial velocity and launch angle.

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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: Calculate the horizontal and vertical components of the initial velocity: u_x = u * cos(ฮธ) and u_y = u * sin(ฮธ). Here, u = 20 m/s and ฮธ = 30 degrees.

\[ u_x = 20 \cos(30^\circ), u_y = 20 \sin(30^\circ) \]

Step 2: Calculate the time of flight using T = (2u_y) / g.

\[ T = \frac{2 \cdot (20 \sin(30^\circ))}{9.81} \]

Step 3: Calculate the range using R = u_x * T.

\[ R = u_x \cdot T \]

A ball is thrown with an initial speed of 15 m/s at an angle of 45 degrees. Find the range of the ball.

Solution:

Step 1: Calculate the horizontal and vertical components of the initial velocity: u_x = 15 * cos(45ยฐ) and u_y = 15 * sin(45ยฐ).

\[ u_x = 15 \cos(45^\circ), u_y = 15 \sin(45^\circ) \]

Step 2: Calculate the time of flight using T = (2u_y) / g.

\[ T = \frac{2 \cdot (15 \sin(45^\circ))}{9.81} \]

Step 3: Calculate the range using R = u_x * T.

\[ R = u_x \cdot T \]

Common Mistakes

  • Mistake: Confusing the launch angle with the angle of elevation when calculating components of velocity.

    Correction: Always ensure to use the correct angle for calculations, which is the angle of launch with respect to the horizontal.

  • Mistake: Neglecting air resistance in calculations, which can lead to inaccuracies in real-world scenarios.

    Correction: Understand that the derived formulas assume a vacuum; for real-world applications, consider air resistance effects.

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