Parametric Equations- Ball travel

In summary, the conversation discusses a baseball being hit 3 feet above the ground and its trajectory path. The table of values shows that the height is equal to 0 at some point, but this is not actually ground level as the origin of the coordinate system is set at 3 feet. The question is whether y=-3 is equal to ground level, which is confirmed by the participants.
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opus
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Suppose a baseball is hit 3 feet above the ground, and that it leaves the bat at a speed of 100 miles an hour at an angle of 20° from the horizontal.

I've got the parametric equations in terms of x and in terms of y, and I have values plotted and a graph sketched. My question is in regards to the initial height of 3 ft, and the position of the ball when it hits the ground. Now in looking at the table of these values, the y value (corresponding to height) is equal to zero at some point in time. Now if one were to look at the table of values, and see that the height is equal to zero feet at some point, is it true that this is not actually 0 ft, since we started from 3 feet? And if we wanted to find out when the ball hit the ground, we'd need to find when the ball was at -3 feet?
 
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  • #2
opus said:
Suppose a baseball is hit 3 feet above the ground, and that it leaves the bat at a speed of 100 miles an hour at an angle of 20° from the horizontal.

I've got the parametric equations in terms of x and in terms of y, and I have values plotted and a graph sketched. My question is in regards to the initial height of 3 ft, and the position of the ball when it hits the ground. Now in looking at the table of these values, the y value (corresponding to height) is equal to zero at some point in time. Now if one were to look at the table of values, and see that the height is equal to zero feet at some point, is it true that this is not actually 0 ft, since we started from 3 feet? And if we wanted to find out when the ball hit the ground, we'd need to find when the ball was at -3 feet?
This all depends on where you selected the origin of the coordinate system (height, width). The resulting parabola is the same, but the equations are different. Theoretically you can also set the origin at 2 ft height and end up with -1 ft, or at even more strange places, e.g. on the score board. However, the feet or the bat of the batter is somehow a natural gauge.
 
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So in this attached image, you can see that we're starting from 3 ft above ground. So as soon as the batter hits the ball, the ball will go through a trajectory path. On it's way back down, it will eventually hit 3 feet above ground level. In terms of the table of values, this would be a height of 0. However, this is clearly not the ground as we started from 3 ft. So, by the table of values, y=-3 is equal to ground level?
 

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  • #4
opus said:
So in this attached image, you can see that we're starting from 3 ft above ground. So as soon as the batter hits the ball, the ball will go through a trajectory path. On it's way back down, it will eventually hit 3 feet above ground level. In terms of the table of values, this would be a height of 0. However, this is clearly not the ground as we started from 3 ft. So, by the table of values, y=-3 is equal to ground level?
Yes. If your horizontal axis (the x-axis) is 3' above ground level, then y = -3 is at ground level.
 
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Thank you both!
 

1. What are parametric equations and how are they used to describe the motion of a ball?

Parametric equations are a set of equations that express the coordinates of a point on a curve in terms of one or more parameters. In the context of ball travel, parametric equations are used to describe the position of the ball at any given time during its motion. This is done by using separate equations for the x and y coordinates, with time as the parameter. By plugging in different values for time, we can track the movement of the ball through the air.

2. How do we determine the initial velocity and angle of a ball using parametric equations?

The initial velocity and angle of a ball can be determined by analyzing the parametric equations for the ball's motion. The initial velocity can be found by looking at the coefficients of the equations for the x and y coordinates, while the initial angle can be determined by finding the ratio between the two equations. Additionally, the initial velocity and angle can also be determined by looking at the initial conditions of the ball's motion, such as the point of release and the direction of the throw.

3. What is the significance of the parametric equations' parameters in relation to the ball's motion?

The parameters in the parametric equations represent the variables that affect the ball's motion. The x and y coordinates are dependent on time, which represents the progression of the ball's movement. Other parameters, such as initial velocity and angle, can also affect the shape and trajectory of the ball's path. By manipulating these parameters, we can adjust the motion of the ball to fit different scenarios.

4. How do we use parametric equations to calculate the maximum height and range of a ball's flight?

The maximum height and range of a ball's flight can be calculated by examining the parametric equations for the ball's motion. The maximum height can be found by looking at the y coordinate at the highest point of the ball's trajectory, which occurs when the derivative of the y equation is equal to 0. The range can be determined by finding the x coordinate at the point where the ball returns to its initial height, which also occurs when the derivative of the y equation is equal to 0.

5. Can parametric equations be used to describe the motion of a ball in three-dimensional space?

Yes, parametric equations can be extended to three-dimensional space to describe the motion of a ball in three dimensions. This would require the addition of a third equation for the z coordinate, with time as the parameter. By using these three parametric equations, we can track the movement of the ball in three-dimensional space and analyze its trajectory and path.

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