Uniform accelerated motion (UAM) equations:

$\overline{)\mathbf{}{{\mathit{v}}}_{{\mathit{f}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathbf{}}{\mathbf{-}}{\mathit{g}}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{y}}{\mathbf{=}}{\mathbf{}}\mathbf{\left(}\frac{{\mathit{v}}_{\mathit{f}}\mathbf{+}{\mathit{v}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{\right)}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{y}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathit{t}}{\mathbf{-}}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{g}}{{\mathit{t}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{}}{{{\mathit{v}}}_{{\mathit{f}}}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{}}{{{\mathit{v}}}_{{\mathbf{0}}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{-}}{\mathbf{2}}{\mathit{g}}{\mathbf{\u2206}}{\mathit{y}}}$

Velocity components:

$\overline{)\begin{array}{rcl}{\mathbf{v}}_{\mathbf{x}}& {\mathbf{=}}& \mathbf{\left|}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{v}}\mathbf{\right|}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{\theta}\\ {\mathbf{v}}_{\mathbf{y}}& {\mathbf{=}}& \mathbf{\left|}\stackrel{\mathbf{\rightharpoonup}}{\mathbf{v}}\mathbf{\right|}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{\theta}\end{array}}$

- v
_{0,y}= v_{0}sin θ = 25 sin θ - v
_{f,y}= 0 m/s - Δy = h
_{max}= 20.4 m - g = 9.8 m/s
^{2}

To understand how the trajectory of an object depends on its initial velocity, and to understand how air resistance affects the trajectory. For this problem, use the PhET simulation Projectile Motion. This simulation allows you to fire an object from a cannon, see its trajectory, and measure its range and hang time (the amount of time in the air). Click to launch video simulation Start the simulation. Press Fire to launch an object. You can choose the object by clicking on one of the objects in the scroll-down menu at top right (a cannonball is not among the choices). To adjust the cannon barrel’s angle, click and drag on it or type in a numerical value (in degrees). You can also adjust the speed, mass, and diameter of the object by typing in values. Clicking Air Resistance displays settings for (1) the drag coefficient and (2) the altitude (which controls the air density). For this tutorial, we will use an altitude of zero (sea level) and let the drag coefficient be automatically set when the object is chosen. Play around with the simulation. When you are done, click Erase and select a baseball prior to beginning Part A. Leave Air Resistance unchecked.

Erase all the trajectories, and fire the ball vertically again with an initial speed of 20 m/s. As you found earlier, the maximum height is roughly 20 m. If the ball isn’t fired vertically, but at an angle less than 90°, it can reach the same maximum height if its initial speed is faster. Set the initial speed to 25 m/s, and find the angle such that the maximum height is roughly 20 m. Experiment by firing the ball with many different angles. You can use the measuring tape to determine the maximum height of the trajectory and compare it to 20 m. What is this angle?