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Analyzing Projectile Motion: Calculating Flight Time, Height, and Range

This guide provides a structured approach to solving projectile motion problems. It outlines four key steps: calculating the X and Y components of the initial velocity using trigonometric functions; determining the maximum height reached; computing hang time and total flight duration; and finally, calculating the range or horizontal distance traveled. Included are practical sample problems involving heights, distances, and angles to enhance understanding. Perfect for students preparing for quizzes or exams in physics or biomechanics.

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Analyzing Projectile Motion: Calculating Flight Time, Height, and Range

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  1. ESS 303 – Biomechanics Projectiles

  2. Which Will Hit First? Y = 1/2 at2

  3. Put Things Together (4 Steps) • Step #1: Calculate the X and Y components of movement. • VX0 = V0 * Cos θ • VY0 = V0 * Sin θ • Step #2: Calculate the maximum height. How far up did it go? • Yup = (VY2 – VY02)/2a • Ydown: How far down will it fall (think about it)?

  4. Put Things Together (4 Steps) • Step #3: Calculate the hang-time • Y = ½at2 • t = √(2Y)/a • Remember to add time up and time down • Step #4: Calculate the range (the horizontal distance) • X = VX0 * t

  5. A Shortcut? v2Sin θCos θ + vCos θ√(vSin θ)2 + 2gh R = –––––––––––––––––––––––––––––––– g

  6. Problems • You drop a penny from the top of a 2000 meter-high building • How long will it take to hit the ground? • How fast will it be going when it hits?

  7. Problems • A player kicked a football giving it a velocity of 20m/s at an angle of 37°. It was caught by another player at a height of 1.5meters. • What were the X and Y components of velocity? • How high did it go? • How long was it in the air? • How far apart were the players?

  8. Quiz 2 Notes • Next week • Bring a calculator (with trig. functions) • Formula sheet provided (same as one on the web) • Scratch paper is recommended • About 16 total points • About 5 problems ranging in points from 2 to 7 • Show your work and include units

  9. Quiz 2 Sample Problems • A runner completes 17 laps of a quarter-mile track. What was his or her distance and displacement? • Distance = 4.25 Miles • Displacement = 0 Miles • Move on a grid from point (3,6) to point (8,10). What are your horizontal, vertical, and resultant displacements? • Horizontal Displacement = 5 • Vertical Displacement = 4 • Resultant Displacement = 6.40

  10. More Quiz 2 Sample Problems • A ballerina twirls 4½ times. What was her angular distance in degrees and radians? • Angular Distance = 1620 Degrees • Angular Distance = 28.27 Radians • While watching a golf swing, you note that the angular velocity at the .18 second mark was 2.5 radians/second. You also see that the angular velocity at the .41 second mark was 12.02 radians/second. What was the angular acceleration? •  = (/t) •  = (12.02-2.5)/.41-.18) = (9.52/0.23) = 41.39 r/s2

  11. 23º YDown = Yup+1.5m Final Quiz 2 Sample Problem • A pitcher throws a ball 1.5 meters above the ground. The ball is released at a 23 degree angle at a velocity of 37 m/s. How far down-field will the ball go (do all 4 steps and show your work)? 37m/s Yup

  12. 23º YDown = Yup+1.5m Final Quiz 2 Sample Problem • Step 1: Horizontal and Vertical Velocities • VX = (V0 * CosӨ) • VX = 34.06 m/s • VY = (V0 * SinӨ) • VY = 14.46 m/s 37m/s Yup

  13. 23º YDown = Yup+1.5m Final Quiz 2 Sample Problem • Step 2: How high did it go (from point of release)? • Y = (Vy02 – Vy12)/(2*a) • Yup = 10.66 meters • Ydown = 12.16 meters 37m/s Yup

  14. 23º YDown = Yup+1.5m Final Quiz 2 Sample Problem • Step 3: What was the flight time? • t = √((2Y)/a) • tup = 1.47 seconds,tdown = 1.57 Seconds • ttotal = 1.47 + 1.57 =3.04 Seconds 37m/s Yup

  15. 23º YDown = Yup+1.5m Final Quiz 2 Sample Problem • Step 4: How far down-field did the ball go? • X = VX * ttotal • X = (34.06 m/s * 3.04 s) =103.54 meters 37m/s Yup

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