How to Calculate Trajectories

Anytime a slugger knocks a ball out of the park or an archer fires an arrow, the object hurtling through the air is following a ballistic path, or trajectory. Determining and predicting this trajectory requires breaking the problem into its horizontal and vertical components. In a ballistic path, acceleration is zero in the horizontal direction, and it is equal to the acceleration of gravity in the vertical direction. Since acceleration is the second derivative of acceleration, integrating these values twice yields the equations for position.

    Break the initial velocity into its vertical and horizontal components. You will already need to know the angle at which the object was fired and its initial velocity. For this example, an archer fires an arrow at 30 degrees with a velocity of 150 ft/sec. V0x = 150_cos(30) = 130 ft/sec V0y = 150_sin(30) = 75 ft/sec

    Choose a value for time and calculate the horizontal distance at that time. It's best to start with zero and work your way through the trajectory incrementally. For this example, the value is calculated at t = 1. x = V0x_t = 130_1 = 130 ft

    Calculate the value for vertical distance at the same time interval. The value for gravitational acceleration in English units is 32.2 ft/sec^2. y = V0y_t - 0.5_g_t^2 = 75(1) - 0.5_32.2*1^2 = 58.9 ft

    Example problem trajectory plot.

    Plot the horizontal and vertical values on a sheet of graph paper. Choose another time value and calculate another set of coordinates. Continue until you have enough points to define your trajectory.


    • The location for y = 0 is wherever the object began free flight, not necessarily the ground. This means an object may travel below zero in the vertical direction on the downward part of the trajectory.


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