Height and Range of a Projectile

• v0x


  Initial velocity along the x-axis
• v0y


  Initial velocity along the y-axis
• θ

  Angle between the initial velocity and the x-axis
• Equations of motion


  1. ᢾᢾ = (ᢾᢾ0 cosᢾᢾ)ᢾᢾ

  2. ᢾᢾ = (ᢾᢾ0sinᢾᢾ)ᢾᢾ −1ᢾᢾᢾᢾ2
• x


  Final position along x-axis
• y


  Final position along y-axis
• v0

  Initial velocity
• vx


  Final velocity along x-axis
• vy

  Final velocity along y-axis
• g

  Acceleration due to gravity (9.81 m/s2)
• The conditions for projectile motion to happen is that an object is thrown with an initial velocity ⃑ᢾᢾ⃑ᢾᢾ` at an angle of launch ᢾᢾ
• The trajectory of the projectile is parabolic
• The peak of the parabola is the highest point where the body could be in its trajectory (y = H)
• At the peak of flight of the projectile ᢾᢾᢾᢾ = ᢾᢾ0sinᢾᢾ − ᢾᢾᢾᢾ = 0
• To find the maximum height H

  1. Substitute ᢾᢾ to the equation of position along the y-axis

  2. ᢾᢾ2sin2ᢾᢾ/2ᢾᢾ = 0
• The time it takes for a projectile to go down to the same level (tdown) is equal to the time it takes when it was rising (tup) to its maximum height
• The total time of flight (T) is just double that time, t
• To find the maximum range (R)

  1. Substitute T to the equation of position along the x-axis

  2. ᢾᢾ2sin 2ᢾᢾ/ᢾᢾ = ᢾᢾ
• When the angle is zero (00), the resulting horizontal displacement is ONLY with no vertical displacement
• When the angle is 900, the resulting horizontal displacement is zero, and the vertical displacement along the y-axis only
• Between 1⁰ and 89⁰, projectile motion is evident given that an initial velocity on the object
• When the angle is 450, the resulting horizontal displacement is maximum with high vertical displacement (but not maximum)
• When the angle is 890, the resulting vertical displacement is maximum with low horizontal displacement (but not minimum)
• Projectile motion is the combination of horizontal motion with constant velocity (acceleration = 0) and vertical motion with constant acceleration (due to gravity)
• The height and range of a projectile are dependent upon 1) initial velocity and 2) angle of launch
• At the maximum height, the vertical component of the velocity is equal to zero
• Before the maximum height, the object is ascending or its height is increasing. After the maximum height, the object is descending or its height is decreasing
• The maximum height reached by a projectile increases as the angle of launch increases until an angle of 90 degrees with respect to the horizontal
• The maximum height reached by a projectile decreases as the angle of launch decreases from 90 degrees
• The range increases as the angle of launch decreases from 45 degrees with respect to the horizontal
• Maximum range is attained at an angle of 45 degrees with respect to the horizontal
• The range decreases as the angle of launch increases from 45 degrees to 90 degrees with respect to the horizontal
• In most cases, increase in initial velocity implies increase in height and range if angle of launch is constant