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solve-problems-on-projectile-motion

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. Projectile motion can be described as a combination of horizontal and vertical motions, which are independent of each other. The path followed by a projectile is a parabola, and the motion can be analyzed using various kinematic equations.

Theory Explanation

Understanding Projectile Motion

Projectile motion occurs when an object is launched into the air and is subject to gravitational acceleration. The motion can be broken down into two components: horizontal motion (constant velocity) and vertical motion (accelerated motion due to gravity). The horizontal motion does not affect the vertical motion and vice versa.

Key Equations of Motion

The key equations used in projectile motion are derived from the basic kinematic equations. For vertical motion, we use: 1. v_y = u_y + at 2. s_y = u_y t + (1/2) a t^2 3. v_y^2 = u_y^2 + 2as_y Where v_y is the final vertical velocity, u_y is the initial vertical velocity, a is the acceleration (which is -g for downward motion), and s_y is the vertical displacement. For horizontal motion, the equations are simpler since there is no acceleration: 1. s_x = u_x t Where s_x is the horizontal displacement and u_x is the horizontal velocity.

Finding Range, Maximum Height, and Time of Flight

The range (R), maximum height (H), and time of flight (T) can be calculated using the following formulas: 1. Range: R = (u^2 sin(2ฮธ)) / g 2. Maximum Height: H = (u^2 sin^2(ฮธ)) / (2g) 3. Time of Flight: T = (2u sin(ฮธ)) / g Where u is the initial velocity, ฮธ is the angle of projection, and g is the acceleration due to gravity (approximately 9.81 m/sยฒ).

Key Points

  • ๐ŸŽฏ Projectile motion is a two-dimensional motion that can be analyzed as two independent one-dimensional motions: horizontal and vertical.
  • ๐ŸŽฏ The horizontal motion has a constant velocity, while the vertical motion is influenced by gravity.
  • ๐ŸŽฏ The path of a projectile is a parabola, and its maximum height, range, and time of flight can be calculated using specific formulas.

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Examples:💡

A ball is thrown with an initial velocity of 20 m/s at an angle of 30 degrees to the horizontal. Calculate the range of the ball.

Solution:

Step 1: First, calculate the horizontal and vertical components of the initial velocity: u_x = u imes cos(ฮธ) = 20 imes cos(30ยฐ) = 20 ร— โˆš3/2 = 17.32 m/s.

\[ u_x = 20 \cos(30ยฐ) = 17.32 m/s \]

Step 2: Next, calculate the time of flight using the formula: T = (2u sin(ฮธ)) / g = (2 ร— 20 ร— sin(30ยฐ)) / 9.81 = (40 ร— 0.5) / 9.81 = 2.04 s.

\[ T = \frac{2 \times 20 \times \sin(30ยฐ)}{9.81} = 2.04 s \]

Step 3: Finally, calculate the range using the formula: R = u_x ร— T = 17.32 ร— 2.04 = 35.32 m.

\[ R = u_x \times T = 17.32 \times 2.04 = 35.32 m \]

A projectile is launched at an angle of 45 degrees with an initial speed of 30 m/s. Find the maximum height reached by the projectile.

Solution:

Step 1: Calculate the vertical component of the initial velocity: u_y = u ร— sin(ฮธ) = 30 ร— sin(45ยฐ) = 30 ร— โˆš2/2 = 21.21 m/s.

\[ u_y = 30 \sin(45ยฐ) = 21.21 m/s \]

Step 2: Use the maximum height formula: H = (u_y^2) / (2g) = (21.21^2) / (2 ร— 9.81) = 225.5041 / 19.62 = 11.51 m.

\[ H = \frac{u_y^2}{2g} = \frac{21.21^2}{2 \times 9.81} = 11.51 m \]

Common Mistakes

  • Mistake: Confusing the horizontal and vertical components of motion, leading to incorrect calculations of range or height.

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

  • Mistake: Neglecting the effect of gravity on the vertical motion, assuming it behaves like horizontal motion.

    Correction: Remember that vertical motion is affected by gravity, which causes acceleration downwards.

  • Mistake: Using incorrect angles in calculations, especially when converting between degrees and radians.

    Correction: Ensure that angles are in the correct unit (degrees or radians) as required by the trigonometric functions.

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