How to Calculate Elevated Archery Shots

At launch, an arrow becomes a single projectile. It is launched at a given angle and accelerated to a constant speed, allowing it to travel through a parabolic path until it reaches its target. Thus, calculating the distance of an elevated archery shot requires calculating the arrow's trajectory. The easiest way to do that is to break the shot into its vertical and horizontal components. Keep in mind that gravity has a significant effect in the path of an arrow, acting as a force during the vertical travel component.

Things You'll Need

  • Stop watch
  • Measuring tape
  • Calculator
  • Blank paper
  • Graph paper
  • Pencil
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Instructions

    • 1

      Determine the arrow's travel velocity and the angle at which it was fired. To calculate the velocity, have an assistant keep track of the time for the arrow to hit the ground, using a stopwatch. Then, measure the distance the arrow traveled. Velocity is distance over time, so once you have the two measurements, you can estimate the arrow's velocity. To estimate the arrow's angle: if you draw the arrow perfectly parallel to the ground the angle will be 0 degrees. If you draw the arrow perfectly perpendicular to the ground, the angle will be 90 degrees. Have your assistant estimate the angle of your shot by using these parameters. If the angle is closer to the parallel line, it will be less than 45 degrees, if it is closer to the perpendicular line, it will be between 45 to 90 degrees.

    • 2

      Calculate the sine and cosine of the shot's angle. Using basic triangle geometry, determine the sine and cosign of the angle. For instance, if you fire an arrow at an angle of 30 degrees, the sine of a 30-degree angle is 0.5; the cosine is 0.866.

    • 3

      Break the arrow's initial velocity (Vo) into its vertical and horizontal components. Continuing with the previous example, if that the arrow's velocity is 120 ft/second:

      Vertical Vo = 120*sine(30 degrees) = 120*0.5 = 60 ft/second

      Horizontal Vo = 120*cos(30 degrees) = 120*0.866 = 104 ft/second

    • 4

      Calculate the vertical distance for a specific time point. The easiest way to do this is to start at time 0 and work your way up. Also, remember that gravity acts as a vertical force on the arrow. The acceleration of gravity is 32 ft/second-squared. Use the following formula to calculate the vertical distance of the example at time = 1:

      Vertical distance = Vertical Vo*t - 0.5*gravity*t-squared
      Vertical distance = 60*1 - 0.5*32*1-squared
      Vertical distance = 60 - 16
      Vertical distance = 44 ft

    • 5

      Calculate the horizontal distance for the same time point. Use the following formula to calculate the horizontal distance of the example at time = 1:

      Horizontal distance = Horizontal Vo*t
      Horizontal distance = 104*1
      Horizontal distance = 104 ft

    • 6

      Using graph paper, plot the vertical and horizontal distance values. Choose different time values to calculate the coordinates of the rest of the arrow's trajectory.