**– Calculate the power needed to function the ski lift.**

**– Calculate the power needed to accelerate this ski lift in 5 s up to the speed of its operation.**

The first objective of this question is to find the **power** required to **operate** the ski lift by first finding the **work** done as the power is equal to the **work done per second**. The power will be calculated with the equation as follows:

\[P=\frac{W}{t}\]

Where W is the work done while t is the time in seconds, the second objective is to find the power required to **accelerate** this ski lift.

This question is based on the application of **Potential and Kinetic energy**. Potential energy is energy that is **stored** and is **dependent** on the relative positions of multiple components of a system. In contrast, Kinetic energy is the object’s energy it has as a result of its **motion**.

## Expert Answer

To calculate the **power** required to lift the ski lift, first, we have to calculate the **work** with the help of the formula:

\[W=mg \Delta z \]

The chairs are separated by $20m$ so at any moment the number of chairs lifted is:

\[N=\frac{1km}{20}=50\]

Next, we have to find the **total mass** with the formula:

\[m=N \times m_ {per chair}=50 \times 250=12500kg\]

\[W=12500 \times 9.81 \times 200 =24525000J\]

In order to calculate the **power** required to operate this ski lift, we first need to calculate the **operation time.**

\[t=\frac{d}{V}=\frac{1km}{10k}=360s\]

Power is defined as the **work done per second**, which is given as:

\[P=\frac{W}{t} = 68125W = 68.125kW\]

Then, we have to calculate the power needed to accelerate this ski lift in $5 s$ up to the speed of its operation.

The lift **acceleration** in 5 sec is:

\[a = \frac {\Delta V}{t}\]

where, $\Delta V$ is the velecoty change.

\[a=10 \times \frac {1000}{3600} – 0\]

\[=0.556 \frac{m}{s^2}\]

The amount of **work** required to **accelerate** the object is equivalent to the change in **kinetic energy** for an object or body and is calculated as:

\[W_a=\frac {1}{2}M(V_2^2 – V_2^1)kJ\]

\[=\frac{1}{2}(12500) \times (7.716)\]

\[=48225.308J\]

\[=48.225 kJ\]

Now the power required to **accelerate** the ski lift in 5s is given as:

\[W_a=\frac {W_a}{\Delta t}kW\]

\[=\frac{48.225}{5}\]

\[=9.645 kW\]

Now calculating the **vertical distance** traveled during the acceleration is given as:

\[h=\frac {1}{2}at^2 sin\propto \]

\[=\frac{1}{2} \times 0.556 \times 5^2 \times \frac{200}{1000}\]

\[=1.39m\]

Now the** power** due to **gravity** is given as:

\[W_g=Mg(z_2 – z_1)\]

\[=\frac {Mgh}{t} \]

\[=\frac {12500 \times 9.81 \times 1.39}{5}\]

\[=34.089 kW\]

Now the **total power** is given as:

\[W_{total}=W_a + W_g\]

\[=9.645 + 34.089\]

\[=43.734 kW \]

## Numeric Result

The **power** required to **operate** the ski lift is $68.125kW$ while the power **required** to **accelerate** this ski lift is $43.734kW$.

## Example

Find the **power** required to **operate** the **ski lift** which operates at a steady **speed** of $10km/h$ and a one-way length of $2km$ with a vertical rise of $300m$ and chairs are spaced $20m$ apart. **Three people** can be seated on each chair with the **average mass** of each loaded chair being $250kg$.

To calculate the **power** required to lift the ski lift, first, we have to calculate the **work** :

\[W=mg \Delta z \]

\[N=\frac{2km}{20}=100\]

Next, we have to find the **total mass,** which is given as:

\[m=N \times m_ {per chair}=100 \times 250=25000kg\]

\[W=25000 \times 9.81 \times 300 =73,575,000J\]

In order to calculate the **power** required to operate this ski lift, we first need to calculate the **operation time.**

\[t=\frac{d}{V}=\frac{2km}{10k}=0.2h=720s\]

Power is defined as the **work done per second**, which is given as:

\[P=\frac{W}{t} = 102187.5W \]