Calculus How To

Problem Solving

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Problem Solving Example: Path of a Baseball

path of a baseball

Image: Cal State LA

The path of a baseball hit by a player is called a parabola. Its graph can be represented in calculus using a pair of parametric functions with time as the dimension. These functions depend on several variables, including:

  • The height from the ground at which the baseball was hit,
  • Its angle of elevation with the horizontal,
  • The initial velocity of the baseball when hit.

Wind speed is another factor that will affect the path of the baseball, but this factor forms complex equations and is not dealt with in these simplified parametric equations.

Path of a baseball: Steps

Step 1: Define the variables used in both the parametric equations.

  • Represent the height in feet by ‘h’,
  • The angle in degrees by ‘a’,
  • The initial velocity in feet per second by ‘v’
  • The time in seconds by ‘t’.

Step 2: Write an equation for the horizontal motion of the baseball as a function of time:

  • x(t) = v * Cos(a) * t.

Step 3: Write an equation to describe the vertical motion of the baseball as a function of time:

  • y (t) = h + v * Sin(a) * t – 16 * t2.

In this formula, t2 is the square of the variable ‘t’, which is simply t * t, or t2.

The pair of x(t) and y(t) equations are the required parametric equations that describe the path of the baseball in calculus.

Tips:

  1. If the initial velocity is known with the unit of miles per hour (mph), it can be converted to the required unit of feet per second (fps) unit. 5280 feet make a mile, 60 minutes make an hour and 60 seconds make a minute. Accordingly, the mph value has to be multiplied by 1.467 to get the fps value.
  2. The value of the trigonometric functions Cos(a) and Sin(a) can be found by using a look-up table or simply by using a calculator. If the ball was hit along the ground, the angle ‘a’ is zero.
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