Question

do the math
calculate energy needs
the equation to calculate the gravitational potential energy (gpe) of an object is gpe = mgh. in this equation, m is the mass of the object in kilograms, g is the acceleration due to earth’s gravity (9.8 m/s²), and h is the height of the object above the ground in meters. energy is expressed in joules (kg · m²/s²).

1. suppose you are designing the launch system for the roller coaster in the diagram. the launch system transfers energy to the cars to begin the ride. if the cars have enough energy to make it over the first hill, the cars can complete the course. the first hill is 61 m higher than the station. the mass of the empty cars is 550.5 kg. the cars can carry up to 12 people, each with a mass of 150 kg. what are the least amounts of energy the launch system must transfer to the cars when they are empty and when they are full to make sure the cars have enough energy to complete the course?

Explanation:

Step1: Find mass of full cars

Calculate total mass of people (\( 12\times150 \)) and add to empty car mass.
\( m_{full}=550.5 + 12\times150=2350.5\,\text{kg} \)

Step2: GPE for empty cars

Use \( \text{GPE}=mgh \) with \( m = 550.5\,\text{kg} \), \( g = 9.8\,\text{m/s}^2 \), \( h = 61\,\text{m} \).
\( \text{GPE}_{empty}=550.5\times9.8\times61 = 329088.9\,\text{J} \)

Step3: GPE for full cars

Use \( \text{GPE}=mgh \) with \( m = 2350.5\,\text{kg} \), \( g = 9.8\,\text{m/s}^2 \), \( h = 61\,\text{m} \).
\( \text{GPE}_{full}=2350.5\times9.8\times61 = 1405128.9\,\text{J} \)

Answer:

When empty: \( 328733.1 \, \text{J} \); When full: \( 1.423\times10^{5} \, \text{J} \) (Wait, correction: When full, let's recalculate properly. Wait, first, let's do step - by - step.

Step1: Calculate the mass of full cars

The mass of empty cars \( m_{empty}=550.5 \, \text{kg} \). Each person has a mass of \( 150 \, \text{kg} \) and there are 12 people. So the mass of people \( m_{people}=12\times150 = 1800 \, \text{kg} \). Then the mass of full cars \( m_{full}=m_{empty}+m_{people}=550.5 + 1800=2350.5 \, \text{kg} \).

Step2: Calculate GPE for empty cars

Using the formula \( \text{GPE}=mgh \), where \( m = m_{empty}=550.5 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), \( h = 61 \, \text{m} \).
\( \text{GPE}_{empty}=550.5\times9.8\times61 \)
First, \( 550.5\times9.8 = 5394.9 \)
Then, \( 5394.9\times61=5394.9\times(60 + 1)=5394.9\times60+5394.9\times1 = 323694+5394.9 = 329088.9 \, \text{J} \) (Wait, earlier miscalculation. Let's do it accurately: \( 550.5\times9.8 = 550.5\times(10 - 0.2)=5505-110.1 = 5394.9 \); \( 5394.9\times61 = 5394.9\times60+5394.9\times1=323694 + 5394.9 = 329088.9 \, \text{J}\approx3.29\times10^{5}\, \text{J}\))

Step3: Calculate GPE for full cars

Using \( \text{GPE}=mgh \), where \( m = m_{full}=2350.5 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), \( h = 61 \, \text{m} \)
\( \text{GPE}_{full}=2350.5\times9.8\times61 \)
First, \( 2350.5\times9.8=(2350 + 0.5)\times9.8=2350\times9.8+0.5\times9.8 = 23030+4.9 = 23034.9 \)
Then, \( 23034.9\times61=23034.9\times(60 + 1)=23034.9\times60+23034.9\times1 = 1382094+23034.9 = 1405128.9 \, \text{J}\approx1.41\times10^{6}\, \text{J}\)

Wait, let's re - check the mass of people: 12 people each with 150 kg: \( 12\times150 = 1800 \, \text{kg} \). Empty car mass: 550.5 kg. So full mass: \( 550.5+1800 = 2350.5 \, \text{kg} \). Correct.

\( g = 9.8 \, \text{m/s}^2 \), \( h = 61 \, \text{m} \).

For empty cars:
\( \text{GPE}=mgh=550.5\times9.8\times61 \)
\( 550.5\times9.8 = 550.5\times9 + 550.5\times0.8=4954.5+440.4 = 5394.9 \)
\( 5394.9\times61 = 5394.9\times60+5394.9\times1 = 323694+5394.9 = 329088.9 \, \text{J}\approx3.29\times10^{5}\, \text{J}\)

For full cars:
\( \text{GPE}=2350.5\times9.8\times61 \)
\( 2350.5\times9.8 = 2350.5\times(10 - 0.2)=23505-470.1 = 23034.9 \)
\( 23034.9\times61 = 23034.9\times60+23034.9\times1 = 1382094+23034.9 = 1405128.9 \, \text{J}\approx1.41\times10^{6}\, \text{J}\)

So the least amount of energy for empty cars is \( 329088.9 \, \text{J} \) (or \( 3.29\times10^{5}\, \text{J} \)) and for full cars is \( 1405128.9 \, \text{J} \) (or \( 1.41\times10^{6}\, \text{J} \)).