Maximum height of a projectile

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SUMMARY

The maximum height (H) of a projectile divided by its range (R) satisfies the relation H/R = 1/4 tan(θ), where θ is the launch angle. To derive this relationship, one must first analyze the projectile's behavior at maximum height and calculate both height and range in terms of initial velocity (v) and angle (θ). Utilizing kinematic equations for horizontal and vertical displacement is essential for establishing the mathematical relationship between these variables.

PREREQUISITES
  • Understanding of kinematic equations for projectile motion
  • Knowledge of trigonometric functions, specifically tangent
  • Familiarity with the concepts of maximum height and range in projectile motion
  • Basic algebra for manipulating equations and ratios
NEXT STEPS
  • Derive the maximum height formula for a projectile using v and θ
  • Calculate the range of a projectile based on initial velocity and launch angle
  • Explore the implications of varying launch angles on H/R ratio
  • Study the effects of air resistance on projectile motion
USEFUL FOR

Students studying physics, educators teaching projectile motion, and anyone interested in the mathematical principles of kinematics.

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Prove that the maximum height of a projectile H, divided by the range of the projectile, R, satisfies the relation H/R = 1/4 tan.


I have no idea how to do this
 
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Why don't you start by figuring out what happens to a projectile at its max heights and then find a mathematical relationship for that. Then do the same for the range, and see what you can come up with.
 
what happens to a projectile at its max heights ?
 
Calculate the height in terms of v which is the speed, and theta which is the angle.

Calculate the range in terms of v and theta.

Take the ratio, and the result follows.

To get the range and height... begin by writing the 2 kinematics equations (that give horizontal and vertical displacement for any time)
 

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