Examples of Projectiles

Examples of Projectiles

Introduction

• When objects are launched into the air with a certain velocity and angle, their motion can be described as projectile motion.
• Understanding key parameters like time of flight, maximum height, and range is essential for analysing such motion.

Given Information

• A girl kicks a ball into the sea at a velocity (v) of 7.2 m/s at an angle (θ) of 30° above the horizontal.

Vertical and Horizontal Components

• To analyse the motion, we must calculate the vertical and horizontal components of the initial velocity.
• The vertical component (v_vertical_initial) is calculated as:
  • v_vertical_initial = v × sin(θ)
• The horizontal component (v_horizontal_initial) is calculated as:
  • v_horizontal_initial = v × cos(θ)

Time of Flight

• The time of flight (t_flight) can be determined using equations of motion since the vertical acceleration is uniform.
• Vertical velocity (v_vertical) at the highest point is 0 m/s, so we can use the equation:
  • v_vertical = v_vertical_initial + a_vertical × t_flight
• Since v_vertical_initial is positive and a_vertical (acceleration due to gravity) is negative, we get:
  • 0 = v_vertical_initial - g × t_flight
• Solving for t_flight, we find:
  • t_flight = (v_vertical_initial)/g
• The total time of flight is then:
  • t_flight_total = 2 × t_flight (as motion is symmetrical)

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Examples of Projectiles

Maximum Height

• To calculate the maximum height (h_max), we need to know the time taken to reach it.
• At the maximum height, the vertical velocity is 0 m/s, so we can use the equation:
  • v_vertical = v_vertical_initial - g × t_to_max
• Solving for t_to_max, we find:
  • t_to_max = (v_vertical_initial)/g
• The maximum height reached is then:
  • h_max = (v_vertical_initial²)/(2 × g)

Range

• Finally, the range (R) is calculated using the horizontal component (v_horizontal_initial) and the total time of flight (t_flight_total):
  • R = v_horizontal_initial × t_flight_total

Calculations

• Given v = 7.2 m/s and θ = 30°, we calculate:
  • v_vertical_initial = 7.2 m/s × sin(30°) = 3.6 m/s
  • v_horizontal_initial = 7.2 m/s × cos(30°) = 6.24 m/s
• Using g = 9.8 m/s²:
  • t_flight = (3.6 m/s)/(9.8 m/s²) ≈ 0.367 s
  • t_flight_total = 2 × 0.367 s = 0.734 s
• Calculating h_max:
  • h_max = (3.6 m/s)²/(2 × 9.8 m/s²) ≈ 0.656 m
• Calculating R:
  • R = 6.24 m/s × 0.734 s ≈ 4.57 m