# If the coaster is going to make it to point C it needs to start with some initial velocity. If...

## Question:

If the coaster is going to make it point C it needs to start with some initial velocity. If point C is 21 meters high and the coaster starts at a height of 20 meters, What would its initial velocity need to be if no energy is lost to friction?

## Application of Conservation of Energy :

We will solve this problem by using the conservation of energy. Since there is no loss in energy due to the friction. The total energy of the coaster will be conserved.

Let the coaster needs to start with minimum speed {eq}v {/eq} so that it can reach {eq}21\ \rm m {/eq} and it has mass {eq}m {/eq}

Hence initial total energy {eq}=\dfrac12mv^2+mgh=\dfrac12mv^2+20mg {/eq}

Since here we are considering minimum speed, therefore, finally, the coaster will have no velocity. It would just barely reach {eq}21\ \rm m {/eq}

Hence final total energy {eq}=\dfrac12mv^2+mgh=0+21mg {/eq}

Now using conservation of energy we get,

\begin{align} &\dfrac12mv^2+20mg=21mg\\[.3 cm] &\dfrac12mv^2=mg\\[.3 cm] &v^2=2g\\[.3 cm] &v=\sqrt{2g}\\[.3 cm] &v=\sqrt{2(9.8\ \rm m/s^2)}\\[.3 cm] &v\approx 4.43\ \rm m/s \end{align}

Hence the coaster need to start with at least velocity of 4.43 m/s. 