A ski lift has a one-way length of 1 km and a vertical rise of 200 m. The chairs

Given parameters:
- Length of ski lift, $L=1km$
- Vertical rise of ski lift, $h=200m$
- Distance between chairs, $d=20m$
- Seating capacity of each chair, $n=3$
- Operating speed of ski lift, $v=10km/h$
- Average mass of each loaded chair, $m=250kg$
- Time to accelerate ski lift, $t=5s$
To find:
- Power required to operate the ski lift, $P_{op}$
- Power required to accelerate the ski lift, $P_{acc}$
First, let's find the time it takes for a chair to travel from the bottom to the top of the ski lift:
$$\begin{gathered} t_c=\frac{L}{v} \\ =\frac{1000m}{(10/3.6)m/s} \\ =360s \end{gathered}$$
The number of chairs on the ski lift can be found as:
$$\begin{gathered} n_c=\frac{L}{d} \\ =\frac{1000m}{20m} \\ =50 \end{gathered}$$
The total mass of the chairs and passengers on the ski lift can be found as:
$$\begin{gathered} m_{tot}=n_c·n·m \\ =50·3·250kg \\ =37500kg \end{gathered}$$
The potential energy of the ski lift at the top can be found as:
$$\begin{gathered} E_{pot}=m_{tot}·g·h \\ =37500kg·9.81m/s^2·200m \\ =7.35·{10}^{7}J \end{gathered}$$
Since the ski lift is operating at a steady speed, the power required to operate it can be found as:
$$\begin{gathered} P_{op}=\frac{E_{pot}}{t_c} \\ =\frac{7.35·{10}^{7}J}{360s} \\ =204166.7W \\ =\boxed{204kW} \end{gathered}$$
To find the power required to accelerate the ski lift, we first need to find the kinetic energy of the ski lift at its operating speed:
$$\begin{gathered} E_{kin}=\frac{1}{2}{m}_{tot}{v}^2 \\ =\frac{1}{2}·37500kg·(10/3.6m/s{)}^2 \\ =2.6·{10}^{7}J \end{gathered}$$
The power required to accelerate the ski lift can be found as:
$$\begin{gathered} P_{acc}=\frac{E_{kin}}{t} \\ =\frac{2.6·{10}^{7}J}{5s} \\ =5.2·{10}^{6}W \\ =\boxed{5200kW} \end{gathered}$$
Therefore, the power required to operate the ski lift is 204 kW, and the power required to accelerate the ski lift in 5 s to its operating speed is 5200 kW.

We can use the following formula to calculate the power required to operate the ski lift:
$P=Fv$
where $P$ is power, $F$ is force, and $v$ is velocity.
To find the force, we first need to find the weight of each chair when it is fully loaded. Assuming an average mass of 75 kg per person, the weight of each loaded chair is:
$W=3(75\text{ kg})+250\text{ kg}=425\text{ kg}$
The force acting on each chair is equal to its weight, which we can find using the formula:
$F=mg$
where $m$ is the mass of the chair and $g$ is the acceleration due to gravity, which is approximately 9.81 m/s${}^2$. Thus, the force on each chair is:
$F=(425\text{ kg})(9.81{\text{ m/s}}^2)=4172.25\text{ N}$
The total force on the lift is equal to the force on each chair multiplied by the number of chairs on the lift:
$F_{total}=(20\text{ m}/\text{chair})(1\text{ km}/20\text{ m})(3\text{ people}/\text{chair})(4172.25\text{ N}/\text{chair})=188252.5\text{ N}$
We can now calculate the power required to operate the ski lift using the formula above:
$P=F_{total}v=(188252.5\text{ N})(10\text{ km/h})(1000\text{ m/km})(1\text{ h}/3600\text{ s})=523.48\text{ kW}$
To find the power required to accelerate the lift to its operating speed in 5 seconds, we can use the formula for acceleration:
$a=\frac{v_f-v_i}{t}$
where $a$ is acceleration, $v_f$ is the final velocity, $v_i$ is the initial velocity, and $t$ is the time taken.
We know that the lift starts from rest, so $v_i=0$. We also know that the lift reaches its operating speed of 10 km/h, which is equal to 2.78 m/s. Thus, the acceleration is:
$a=\frac{2.78\text{ m/s}-0}{5\text{ s}}=0.556{\text{ m/s}}^2$
The force required to accelerate the lift is equal to its mass multiplied by the acceleration:
$F=ma$
We can estimate the mass of the lift by assuming that each chair is evenly spaced along the lift, so the lift consists of 50 chairs:
$m_{lift}=50(425\text{ kg})=21250\text{ kg}$
Thus, the force required to accelerate the lift is:
$F=(21250\text{ kg})(0.556{\text{ m/s}}^2)=11818.75\text{ N}$
Finally, we can calculate the power required to accelerate the lift using the formula above:
$P=Fv=(11818.75\text{ N})(2.78\text{ m/s})=32833.13\text{ W}=32.83\text{ kW}$
Therefore, the power required to operate the ski lift is $523.48$ kW and the power required to accelerate is $32.83\text{ kW}$

Answer:
2951 W.
Explanation:
Given:
Length of ski lift, $d=1$ km $=1000$ m.
Speed of ski lift, $v=10$ km/h $=\frac{10}{3.6}$ m/s.
The time taken to travel the length of the ski lift is given by:
$t=\frac{d}{v}.$
Substituting the given values:
$t=\frac{1000}{\frac{10}{3.6}}=\frac{1000\times 3.6}{10}=360\text{ s}.$
Next, let's determine the number of chairs on the ski lift.
Given:
Distance between chairs, $l=20$ m.
Length of ski lift, $d=1000$ m.
The number of chairs on the ski lift is given by:
$n=\frac{d}{l}.$
Substituting the given values:
$n=\frac{1000}{20}=50.$
Now, let's determine the work done in lifting the chairs.
Given:
Vertical rise, $h=200$ m.
Mass of loaded chair, $m=250$ kg.
Acceleration due to gravity, $g=9.8$ m/s${}^2$.
The work done in lifting one chair is given by:
$W=mgh.$
Substituting the given values:
$W=250\times 9.8\times 200=490,000\text{ J}.$
Since each chair can seat three people, the work done in lifting three people is three times the work done for one chair:
$W_{\text{total}}=3W=3\times 490,000=1,470,000\text{ J}.$
The power required to operate the ski lift is given by the work done divided by the time taken:
$P=\frac{W_{\text{total}}}{t}.$
Substituting the values we calculated earlier:
$P=\frac{1,470,000}{360}\approx 4,083\text{ W}.$
Therefore, the power required to operate this ski lift is approximately 4083 W.
To estimate the power required to accelerate the ski lift in 5 seconds to its operating speed when it is first turned on, we can calculate the change in kinetic energy and divide it by the time taken.
The initial speed, $v_0=0$ m/s.
The final speed, $v=10$ km/h $=\frac{10}{3.6}$ m/s.
The change in kinetic energy is given by:
$\Delta KE=\frac{1}{2}m{v}^2-\frac{1}{2}m(v_0{)}^2.$
Substituting the given values:
$\Delta KE=\frac{1}{2}\times 50\times 250{\left(\frac{10}{3.6}\right)}^2-\frac{1}{2}\times 50\times 250(0{)}^2.$
Simplifying the equation:
$\Delta KE=\frac{1}{2}\times 50\times 250\times {\left(\frac{10}{3.6}\right)}^2.$
Next, we can calculate the power using the equation:
$P=\frac{\Delta KE}{\Delta t}.$
Given $\Delta t=5$ s, we can substitute the values:
$P=\frac{\frac{1}{2}\times 50\times 250\times {\left(\frac{10}{3.6}\right)}^2}{5}.$
Calculating the power:
$P\approx 2,951\text{ W}.$
Therefore, the power required to accelerate this ski lift in 5 seconds to its operating speed when it is first turned on is approximately 2951 W.