# University of Wisconsin Green Bay

A major league baseball player hits a fly ball. The ball leaves the bat with a velocity of 41 m/s at an angle of 72o above the horizontal. How far from home plate does the center fielder need to stand if he is going to catch the ball at the same height from which it was hit?

• In this problem, you are asked to describe the motion (how far it travels before it returns to its original height) of a baseball. Whenever you are asked to describe the motion of an object without worrying about the cause of that motion, you have a kinematics problem.

• There are three key kinematic equations. If you carefully select the equation which most directly describes the situation in your problem, you will not only solve the problem in fewer steps but also understand it better. The three equations, written for motion in the y-direction, are:

1. y = y0 + v0yΔt + ½ ay(Δt)2 (relates position and time)
2. vy = v0y + ayΔt (relates velocity and time)
3. vy2 = v0y2 + 2ay(Δy) (relates velocity and position)

In this problem, you are asked where to stand to catch the ball (x-position of the ball) when it returns to its original height (y-position) of the ball. But position is a vector—you cannot mix x- and y-components in the same equation.

Whenever you are asked about the x-direction and told about the y-direction, or vice versa, you need to use time to go between the two.

The amount of time that the baseball spends going from the original height to the final height is the same as the amount of time it takes to go from the bat to the center fielder. So you can use what you know about the y-position of the ball to find the time it spends in the air (equation 1 in the y-direction) and then use that time to find its x-position (equation 1 in the x-direction.)

• Step 1:

Equation 1 will always give two solutions for time. In this case, t = 0 corresponds to the time that the ball left the bat, and t = 7.96 s corresponds to the time that it returned to that same height in center field. Now that you know how long the ball was in the air, proceed to step 2 to find how far it traveled.

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Step 2:

The problem asked how far the base ball traveled from home plate. No further mathematical solution is necessary.

• In this problem, you were asked to find how far a baseball travels during the time it spends going up and coming down to the same height. Displacement, velocity and acceleration are vector quantities, and so you cannot mix x- and y- components in the same equation. Therefore, you cannot use the y-position condition (same height) to solve for the x-position question (where to stand to catch the ball) in a single step.

You can always relate x- and y-motion through time—the ball spends as long going up and down as it spends going from home plate to center field. So we first solved for time required to go up and down (y-position) and then used that time to determine how far the ball traveled (x-position.)