<div class="ns-school-emp">
<h2>Example: The Roller Coaster</h2>
<div class="ns-school-defn">
<h3>Question</h3>
<div>
<p>A roller coaster ride at an amusement park starts from rest at a height of $\text{50}$ $\text{m}$ above the ground and rapidly drops down along its track. At some point, the track does a full 360° loop which has a height of $\text{20}$ $\text{m}$, before finishing off at ground level. The roller coaster train itself with a full load of people on it has a mass of $\text{850}$ $\text{kg}$.</p>
<h3><strong>Roller Coaster</strong></h3>
<img alt="" src="https://data.ulearngo.com/assets/c1983a3e-54b1-4d48-97d7-f33749df12e7"/>
<p>If the roller coaster and its track are frictionless, calculate:</p>
<ol>
<li>
<p>the velocity of the roller coaster when it reaches the top of the loop</p>
</li>
<li>
<p>the velocity of the roller coaster at the bottom of the loop (i.e. ground level)</p>
</li>
</ol>
</div>
<div>
<h3>Step 1: Analyse the question to determine what information is provided</h3>
<ul>
<li>
<p>The mass of the roller coaster is $m = \text{850}\text{ kg}$</p>
</li>
<li>
<p>The initial height of the roller coaster at its starting position is ${h}_{1} = \text{50}\text{ m}$</p>
</li>
<li>
<p>The roller coaster starts from rest, so its initial velocity ${v}_{1} = \text{0}\text{ m·s$^{-1}$}$</p>
</li>
<li>
<p>The height of the loop is ${h}_{2} = \text{20}\text{ m}$</p>
</li>
<li>
<p>The height at the bottom of the loop is at ground level, ${h}_{3} = \text{0}\text{ m}$</p>
</li>
</ul>
<p>We do not need to convert units as they are in the correct form already.</p>
</div>
<div>
<h3>Step 2: Analyse the question to determine what is being asked</h3>
<ul>
<li>
<p>the velocity of the roller coaster at the top of the loop</p>
</li>
<li>
<p>the velocity of the roller coaster at the bottom of the loop</p>
</li>
</ul>
</div>
<div>
<h3>Step 3: Calculate the velocity at the top of the loop</h3>
<p>From the conservation of mechanical energy, We know that at any two points in the system, the total mechanical energy must be the same. Let's compare the situation at the start of the roller coaster to the situation at the top of the loop:</p>

\begin{align*} {E}_{M1} &amp; = {E}_{M2} \\ {E}_{K1} + {E}_{P1} &amp; = {E}_{K2} + {E}_{P2} \\ 0 + mg{h}_{1} &amp; = \frac{1}{2}m{\left({v}_{2}\right)}^{2} + mg{h}_{2} \end{align*}

<p>We can eliminate the mass, $m$, from the equation by dividing both sides by $m$.</p>

\begin{align*} g{h}_{1} &amp; = \frac{1}{2}{\left({v}_{2}\right)}^{2} + g{h}_{2} \\ {\left({v}_{2}\right)}^{2} &amp; = 2\left(g{h}_{1} - g{h}_{2}\right) \\ {\left({v}_{2}\right)}^{2} &amp; = 2\left(\left(\text{9.8}\text{ m·s$^{-2}$}\right)\left(\text{50}\text{ m}\right)-\left(\text{9.8}\text{ m·s$^{-2}$}\right)\left(\text{20}\text{ m}\right)\right) \\ {v}_{2} &amp; = \text{24.25}\text{ m·s$^{-1}$} \end{align*}</div>
<div>
<h3>Step 4: Calculate the velocity at the bottom of the loop</h3>
<p>Again we can use the conservation of energy and the total mechanical energy at the bottom of the loop should be the same as the total mechanical energy of the system at any other position. Let's compare the situations at the start of the roller coaster's trip and the bottom of the loop:</p>

\begin{align*} {E}_{M1} &amp; = {E}_{M3} \\ {E}_{K1} + {E}_{P1} &amp; = {E}_{K3} + {E}_{P3} \\ \frac{1}{2}{m}_{1}{\left(0\right)}^{2} + mg{h}_{1} &amp; = \frac{1}{2}m{\left({v}_{3}\right)}^{2} + mg\left(0\right) \\ mg{h}_{1} &amp; = \frac{1}{2}m{\left({v}_{3}\right)}^{2} \\ {\left({v}_{3}\right)}^{2} &amp; = 2g{h}_{1} \\ {\left({v}_{3}\right)}^{2} &amp; = 2\left(\text{9.8}\text{ m·s$^{-2}$}\right)\left(\text{50}\text{ m}\right) \\ {v}_{3} &amp; = \text{31.30}\text{ m·s$^{-1}$} \end{align*}</div>
</div>
</div>

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

Roller Coaster Example

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

Explore the concepts of mechanical energy, potential and kinetic energy, work, power, and conservation of energy through examples like pendulum, roller coaster, and inclined plane.


Roller Coaster Example

Track Your Learning Progress