Question

Let \(v(t)\) and \(w(t)\) be the horizontal and vertical components of the velocity of a batted (or thrown) bascball. In the absence of air resistance, \(v\) and \(w\) satisfy the equations $$ d v / d t=0, \quad d w / d t=-g $$ $$ \text { (a) Show that } $$ $$ \mathbf{v}=u \cos A, \quad w=-g t+u \sin A $$ $$ \begin{array}{l}{\text { where } u \text { is the initial speed of the ball and } A \text { is its initial angle of elevation, }} \\ {\text { (b) Let } x(t) \text { and } y(t), \text { respectively, be the horizontal and vertical coordinates of the ball at }} \\ {\text { time } t \text { . If } x(0)=0 \text { and } y(0)=h, \text { find } x(t) \text { and } y(t) \text { at any time } t} \\ {\text { (c) Let } g=32 \text { flsec', } u=125 \mathrm{ft} / \mathrm{sec}, \text { and } h=3 \mathrm{ft} \text { . Plot the trajectory of the ball for }} \\ {\text { several values of the angle } A, \text { that is, plot } x(t) \text { and } y(t) \text { parametrically. }} \\ {\text { (d) Suppose the outfield wall is at a distance } L \text { and has height } H \text { . Find a relation between }}\end{array} $$ $$ \begin{array}{l}{u \text { and } A \text { that must be satisfied if the ball is to clear the wall. }} \\ {\text { (e) Suppose that } L=350 \mathrm{ft} \text { and } H=10 \mathrm{ft} \text { . Using the relation in part (d), find (or estimate }} \\ {\text { from a plot) the range of values of } A \text { that correspond to an initial velocity of } u=110 \mathrm{ft} \text { sec. }} \\\ {\text { (f) For } L=350 \text { and } H=10 \text { find the minimum initial velocity } u \text { and the corresponding }} \\ {\text { optimal angle } A \text { for which the ball will clear the wall. }}\end{array} $$

Short answer

Short Answer: The desired relation between initial velocity u and angle A is $$ u\sin A = \frac{1}{2}g\frac{t}{u\cos A} + \frac{H}{L}u\cos A - \frac{h}{L}u\cos A $$. For the given parameters, the range of values for angle A can be estimated from a plot, and the minimum initial velocity u and its corresponding optimal angle A can be found using optimization techniques or computational methods.

Step-by-step solution

(a) Velocity equations derivation

Start with the given differential equations: $$ \frac{dv}{dt} = 0, \qquad \frac{dw}{dt}=-g $$ Integrate the equations with respect to time, t: $$ \int{dv} = \int{0 dt} \Rightarrow v = u\cos A + C_1 $$ $$ \int{dw} = \int{-g dt} \Rightarrow w = -gt + C_2 $$ As the initial values are v(0) = u*cos(A) and w(0) = u*sin(A), the constants C1 and C2 are equal to u*cos(A) and u*sin(A), respectively. Therefore, we obtain: $$ v = u\cos A, \quad w = -gt + u\sin A $$

(b) Position equations derivation

We are given the horizontal and vertical motion equations: $$ x(t) = \int{v dt}, \quad y(t) = \int{w dt} $$ Integrate each equation with respect to time, t: $$ \int{x'(t) dt} = \int{u\cos A dt} \Rightarrow x(t) = u\cos A t + C_3 $$ $$ \int{y'(t) dt} = \int{(-gt + u\sin A)dt} \Rightarrow y(t) = -\frac{1}{2}gt^2 + u\sin A t + C_4 $$ Using the initial conditions x(0) = 0 and y(0) = h, we find the constants C3 and C4: $$ C_3 = 0, \quad C_4 = h $$ Thus, the position equations at any time t are: $$ x(t) = u\cos A t, \quad y(t) = -\frac{1}{2}gt^2 + u\sin A t + h $$

(c) Plot trajectory

The student will need to use a graphing calculator or a computer graphing software with given values of g, u, and h to plot x(t) and y(t) parametrically for several values of A.

(d) Relation between u and A

To clear the wall, the ball must have coordinates (L, H) at some time t. Plug x = L and y = H into the position equations: $$ L = u\cos A t $$ $$ H = -\frac{1}{2}gt^2 + u\sin A t + h $$ Divide second equation by the first equation to eliminate t: $$ \frac{H}{L} = -\frac{1}{2}g\frac{t^2}{t} + \frac{u}{u\cos A} \sin A + \frac{h}{L} $$ Simplify for desired relation: $$ u\sin A = \frac{1}{2}g\frac{t}{u\cos A} + \frac{H}{L}u\cos A - \frac{h}{L}u\cos A $$

(e) Range of A with u = 110 ft/sec

Use the derived relation from part (d) and insert the values L=350 ft, H=10 ft, and u=110 ft/sec. To find the range of values for angle A, plot the relation and estimate the values from the plot.

(f) Minimum initial velocity and corresponding optimal angle

To minimize the initial speed, we should find the minimum value of u that satisfies the relation in part (d) for the given values of L and H. Use optimization techniques or computational methods to find the minimum value of u and the corresponding optimal angle A.

Key concepts

Differential Equations

Differential equations are equations that relate a function with its derivatives. They play a crucial role in modeling physical phenomena, such as projectile motion. In the exercise, we have two differential equations for a baseball: \( \frac{dv}{dt} = 0 \) and \( \frac{dw}{dt} = -g \). These equations describe how the horizontal and vertical components of velocity change over time in the absence of air resistance.

• Horizontal Motion: The equation \( \frac{dv}{dt} = 0 \) tells us that the horizontal velocity \(v\) is constant over time since there's no acceleration involved.
• Vertical Motion: The equation \( \frac{dw}{dt} = -g \) highlights that the vertical component \(w\) is influenced by gravity, with \(g\) being the acceleration due to gravity.

By integrating these equations with respect to time, we determine the functions for both velocity components, resulting in \(v = u \cos A\) and \(w = -gt + u \sin A\). These equations express the velocity components in terms of time \(t\), initial speed \(u\), and angle \(A\).

Projectile Trajectories

Understanding projectile trajectories involves analyzing the path a projectile follows under the influence of gravity. The trajectory of an object can be described using its position equations for both horizontal and vertical movements. In our exercise, we derived these using initial conditions, yielding:

• Horizontal Position: \(x(t) = u \cos A \cdot t\)
• Vertical Position: \(y(t) = -\frac{1}{2}g t^2 + u \sin A \cdot t + h\)

These equations show how the projectile moves in both dimensions over time. The initial height \(h\), alongside the components influenced by initial velocity and angle of projection, fully define its path.

To visualize how different launch angles \(A\) affect the projectile’s trajectory, students can plot these equations parametrically, examining paths under varying angles.

Optimal Angle Calculation

When determining how to clear obstacles, like an outfield wall, in projectile motion, optimal angle calculations are essential. The objective is to find the angle \(A\) that allows the projectile to reach over a given wall height while traveling a specific distance \(L\).
Relation for Clearing the Wall:
To establish a relationship between initial speed \(u\) and angle \(A\), we substitute \(x = L\) and \(y = H\) into motion equations, simplifying to understand required velocity components.

The key equation derived:
\[u \sin A = \frac{1}{2} g\frac{t}{u \cos A} + \frac{H}{L}u\cos A - \frac{h}{L}u\cos A\]
This helps evaluate the optimal angles for clearing specific heights at different distances.
Setting values specific to the problem (like \(L = 350 \text{ ft}\) and \(H = 10 \text{ ft}\)), a plot or calculation allows for estimation of appropriate angle values, revealing the required launch configurations.

Parametric Equations

Parametric equations allow us to express the coordinates of a moving object as functions of an independent parameter, in this case, time \(t\). This is particularly useful in analyzing projectile motion where both the x and y coordinates change dynamically with time.

For the baseball example, the parametric equations are:

• \(x(t) = u \cos A \cdot t\)
• \(y(t) = -\frac{1}{2} g t^2 + u \sin A \cdot t + h\)

These equations give us a continuous representation of the trajectory of the projectile.

By plotting these parametric equations for different angles \(A\), students can gain insights into how varying the angle impacts the baseball's path, helping them grasp the nuances of projectile motion in practical scenarios.