Question

Use the model for projectile motion, assuming there is no air resistance. Rogers Centre in Toronto, Ontario has a center field fence that is 10 feet high and 400 feet from home plate. A ball is hit 3 feet above the ground and leaves the bat at a speed of 100 miles per hour. (a) The ball leaves the bat at an angle of \(\theta=\theta_{0}\) with the horizontal. Write the vector-valued function for the path of the ball. (b) Use a graphing utility to graph the vector-valued function for \(\theta_{0}=10^{\circ}, \theta_{0}=15^{\circ}, \theta_{0}=20^{\circ},\) and \(\theta_{0}=25^{\circ} .\) Use the graphs to approximate the minimum angle required for the hit to be a home run. (c) Determine analytically the minimum angle required for the hit to be a home run.

Short answer

With the initial horizontal position of the ball being 0, the velocity vector at the angle \(\theta\) is given by \(<100cos(\theta), 100sin(\theta)>\). The position vector as a function of time will then be \(<100tcos(\theta), 100tsin(\theta) - 16t^{2} + 3>\). The graph of this function can give a rough idea of the angle needed, but the exact minimum angle that will clear the fence can be found using the analytical approach, which involves setting the distribution equal to the height of the wall and solving for \(\theta\).

Step-by-step solution

Convert measurements

Firstly, convert all measurements into a uniform system. Typically in physics, the SI system is applied, hence the distance to the fence and its height are converted from feet to meters, and the initial speed from miles per hour to meters per second.

Parameterize the velocity vector

Given the magnitude of the initial velocity and the launch angle, a velocity vector can be formed. Use \(v_{0x}=v_{0} \cos(\theta)\) and \(v_{0y}=v_{0} \sin(\theta)\) to compute the x and y components of the velocity vector.

Derive the position vector function

Use the equations for motion in the horizontal and vertical directions, \(x(t)=v_{0x}t\) and \(y(t)=v_{0y}t - \frac{1}{2}gt^{2}\), to form the function, \(\), for the path of the ball.

Graph the function

Use a graphing tool to plot the function for the angles mentioned. These plots will give a visual representation of the trajectory of the ball for different launch angles.

Calculate minimum launch angle for a home run

Analyze the time \(t_{f}\) it takes for the ball to reach the fence, which can be found from the x-component of the path function equal to the distance to the fence. Substitute this time into the y-component of the path function, and set this equal to the height of the wall. This equation can be solved for \(\theta\) to find the minimum angle required to clear the fence.

Key concepts

Vector-Valued Function

In physics and mathematics, the concept of a vector-valued function is a foundational tool. Unlike typical functions that output a single value, vector-valued functions output a vector, which means they provide multiple pieces of information simultaneously. For instance, in projectile motion, a vector-valued function may represent the position of an object in three-dimensional space as a function of time.

The function is typically written as \(R(t) = \langle x(t), y(t), z(t) \rangle\), where \(R(t)\) is the vector-valued function, and \(x(t)\), \(y(t)\), and \(z(t)\) are the scalar functions describing the movement of the object along the x, y, and z-axis, respectively. In the case of a projectile like a baseball, we usually assume a flat, two-dimensional plane, disregarding the z-axis.

Returning to the context of our baseball example, the function for the path of the ball would consider only the horizontal distance (x) and vertical height (y) from the perspective of the initial position, thus the z-component is omitted. The vector-valued function gives us a comprehensive view of the projectile's trajectory over time, depicting both its height and distance traveled concurrently.

Parametric Equations in Calculus

When we transition from a scalar to a vector-based context to discuss motion, we often rely on parametric equations. These equations delineate the coordinates of a point on the curve as a function of a third variable, typically time (t). In calculus, parametric equations are crucial for modeling and analyzing motion where each coordinate is a function of time rather than directly of each other.

For a projectile in motion without air resistance, its horizontal and vertical positions are defined by two separate equations: \(x(t) = v_{0x}t\) and \(y(t) = v_{0y}t - \frac{1}{2}gt^2\), wherein \(v_{0x}\) and \(v_{0y}\) are the initial velocity components in the x and y directions, respectively, and \(g\) is the acceleration due to gravity. Time (\(t\)) acts as the parameter linking the two equations.

These parametric equations give us an analytical tool to explore how the projectile's horizontal displacement and height evolve over time. This piece of the puzzle is vital when solving for unknowns in the motion, such as the angle needed for a home run.

Analytical Geometry

The realm of analytical geometry, also referred to as coordinate geometry, is where algebra meets geometry, allowing complex spatial problems to be solved through algebraic equations. Here, the concepts of lines, curves, and shapes are converted into a numeric framework, making it easier to analyze and calculate their properties.

In the context of a baseball trajectory, analytical geometry enables us to establish the path's equation and solve for variable characteristics like the angle or the exact point where the ball crosses the outfield fence, which is a direct application of this discipline.

By setting the horizontal component of the ball's displacement equal to the distance from home plate to the fence and the vertical component to the height of the fence, we engage in analytical geometry. This approach allows us to deduce the minimum launch angle required for the ball to be considered a home run by finding the point where the ball's trajectory and the fence intersect.

Kinematic Equations

The kinematic equations describe the motion of objects without regard to the forces causing the motion. They are a set of formulas that predict the future position and velocity of an object moving under constant acceleration.

Among the set of kinematic equations, two are particularly relevant for projectile motion: \(x(t) = v_{0x}t\) and \(y(t) = v_{0y}t - \frac{1}{2}gt^2\). These provide the linkage between time and spatial displacement, which can help us work out how far and how high the projectile will go.

When applying these equations to solve for the minimum angle for a home run, you calculate the time it takes for the ball to reach the 400 feet distance to the fence and then use this time to find out if the ball's height at that time would be above 10 feet. These calculations are central to understanding the conditions under which the baseball would successfully fly over the fence, thereby achieving a home run.