<p>The following example illustrates how to mathematically describe projectile motion.</p>
<div class="ns-school-emp">
<h2>Example: Describing Projectile Motion</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A cricketer hits a cricket ball from the ground and the following graph of velocity vs time was drawn. Upwards was taken as positive. Study the graph and follow the instructions below:</p>
<ol>
<li>
<p>Describe the motion of the ball according to the graph.</p>
</li>
<li>
<p>Draw a sketch graph of the corresponding position-time graph. Label the axes.</p>
</li>
<li>
<p>Draw a sketch graph of the corresponding acceleration-time graph. Label the axes.</p>
</li>
</ol>
<img alt="23c40668c98b794a8294a11cee2f3493.png" src="https://data.ulearngo.com/assets/04bb44a1-612e-4132-9992-0eaeff81f837"/></div>
<div>
<h3>Step 1: Describe the motion of the ball</h3>
<p>We need to study the velocity-time graph to answer this question. We will break the motion of the ball up into two time zones: $t=\text{0}\text{ s}$ to $t=\text{2}\text{ s}$ and $t=\text{2}\text{ s}$ to $t=\text{4}\text{ s}$.</p>
<p>From $t=\text{0}\text{ s}$ to $t=\text{2}\text{ s}$ the following happens:</p>
<p>The ball starts to move at an initial velocity of $\text{19.6}$ $\text{m·s $^{-1}$}$ and decreases its velocity to $\text{0}$ $\text{m·s $^{-1}$}$ at $t=\text{2}\text{ s}$. At $t=\text{2}\text{ s}$ the velocity of the ball is $\text{0}$ $\text{m·s $^{-1}$}$ and therefore it stops.</p>
<p>From $t=\text{2}\text{ s}$ to $t=\text{4}\text{ s}$ the following happens:</p>
<p>The ball moves from a velocity of $\text{0}$ $\text{m·s $^{-1}$}$ to $\text{19.6}$ $\text{m·s $^{-1}$}$ in the opposite direction to the original motion.</p>
<p>If we assume that the ball is hit straight up in the air (and we take upwards as positive), it reaches its maximum height at $t= \text{2}\text{ s}$, stops, and falls back to the Earth to reach the ground at $t= \text{4}\text{ s}$.</p>
</div>
<div>
<h3>Step 2: Draw the position-time graph</h3>
<p>To draw this graph, we need to determine the displacements at $t = \text{2}\text{ s}$ and $t = \text{4}\text{ s}$.</p>
<p>At $t = \text{2}\text{ s}$:</p>
<p>The displacement is equal to the area under the graph:</p>
<p>Area under graph = Area of triangle</p>
<p>$\text{Area}=\frac{1}{2}bh$</p>
<p>$\text{Area}=\frac{1}{2}\times 2\times \text{19.6}$</p>
<p>Position = $\text{19.6}$ $\text{m}$</p>
<p>At $t = \text{4}\text{ s}$:</p>
<p>The displacement is equal to the area under the whole graph (top and bottom). Remember that an area under the time line must be subtracted:</p>
<p>Area under graph = Area of triangle 1 + Area of triangle 2</p>
<p>$\text{Area}=\frac{1}{2}bh+\frac{1}{2}bh$</p>
<p>$\text{Area}=\left(\frac{1}{2}\times 2\times \text{19.6}\right)+\left(\frac{1}{2}\times 2\times \left( -\text{19.6}\right)\right)$</p>
<p>$\text{Area}=19.6-19.6$</p>
<p>Displacement = $\text{0}$ $\text{m}$</p>
<p>The position vs time graph for motion at constant acceleration is a curve. The graph will look like this:</p>
<img alt="0c3125c54ce82a00b652654ed8a5a115.png" src="https://data.ulearngo.com/assets/f5657daf-1a35-438c-a2a2-b65693d520ca"/></div>
<div>
<h3>Step 3: Draw the acceleration versus time graph</h3>
<p>To draw the acceleration vs time graph, we need to know what the acceleration is. The velocity versus time graph is a straight line which means that the acceleration is constant. The gradient of the line will give the acceleration.</p>
<p>The line has a negative slope (goes down towards the left) which means that the acceleration has a negative value.</p>
<p>Calculate the gradient of the line:</p>

\begin{align*} \text{gradient}=\frac{\Delta \vec{v}}{\Delta t} \\ \text{gradient}=\frac{0 -\text{19,6}}{2-0} \\ \text{gradient}=\frac{ -\text{19,6}}{2} \\ \text{gradient}= -\text{9,8} \end{align*}

<p>Therefore the acceleration = $\text{9.8}$ $\text{m·s $^{-2}$}$ downwards.</p>
<img alt="fba66049d8ab488fdebb327945baf523.png" src="https://data.ulearngo.com/assets/53e19b8c-9d2f-4878-a578-1ca0b7c92e80"/></div>
</div>
</div>

1c4ddb66-ad36-4f81-90e7-4629f497d0b4

Describing Projectile Motion

Physics is a science discipline concerned with the nature and properties of matter and energy. Physics topics include mechanics, heat, light and radiation, sound, electricity, magnetism, the structure of atoms, and so on.

Physics

Learn about the basic principles of motion including reference frames, position, displacement, distance, speed, velocity, acceleration, stationary objects, graphs of motion, projectile motion, and equations of motion with practical examples and applications.


Describing Projectile Motion

Track Your Learning Progress