When engineers design a roller coaster, they primarily use the principle of conservation of mechanical energy. The motor at the beginning does work to pull the train to the top of the first, highest hill, giving it maximum gravitational potential energy ($U_g = mgh$). As it drops, this potential energy is converted into kinetic energy ($K = \frac{1}{2}mv^2$), determining its maximum speed at the bottom. The coaster can never climb a subsequent hill higher than the first one unless a motor does more work, because friction and air resistance (non-conservative forces) continuously dissipate mechanical energy as heat and sound throughout the ride.

A hydroelectric dam converts the gravitational potential energy of water stored in a high reservoir into electrical energy. As water falls through the penstock, its potential energy becomes kinetic energy, which spins a turbine (mechanical work). The turbine spins a generator to produce electricity. Engineers must calculate the available power $P = \frac{mgh}{t} = \rho Q g h$, where $Q$ is the volumetric flow rate. However, real systems are never 100% efficient due to friction in the pipes, mechanical losses in the turbine, and electrical losses in the generator. Therefore, the actual electrical power output is always less than the theoretical fluid power available.

A worker pushes a 50 kg crate a distance of 8 meters across a level floor with a constant horizontal force of 120 N. How much work does the worker do on the crate?

An ideal spring with a spring constant $k = 400 \text{ N/m}$ is compressed by $0.15 \text{ m}$ from its equilibrium position. How much work is done by the spring as it expands back to equilibrium ($x=0$)?

A particle is subjected to a force $F(x) = (3x^2 - 2x + 5) \text{ N}$. Calculate the work done by this force as the particle moves from $x = 1 \text{ m}$ to $x = 3 \text{ m}$.

A 2 kg ball is dropped from rest from a height of 10 meters. Ignore air resistance. What is its velocity just before it hits the ground? ($g = 9.8 \text{ m/s}^2$)

A roller coaster car (mass $m$) starts from rest at the top of a hill of height $h$. It then enters a circular loop of radius $R$. Assuming no friction, what is the minimum height $h$ required so that the car doesn't fall off the track at the very top of the loop?

A 5 kg block slides down a $30^\circ$ incline that is 4 meters long. It starts from rest at the top. The coefficient of kinetic friction is $\mu_k = 0.25$. Use energy methods to find its speed at the bottom of the incline.

An electric motor lifts a 1500 kg elevator cab 20 meters upward at a constant speed in 12 seconds. What is the average power output of the motor in Watts and Horsepower? ($1 \text{ hp} \approx 746 \text{ W}$)

A car engine applies a constant forward thrust force of 3000 N to maintain a steady highway speed of $25 \text{ m/s}$ (approx 90 km/h) against air resistance and friction. What is the instantaneous power output of the engine?

A water pump is rated at $5.0 \text{ kW}$ of electrical power input. It is used to pump water from a well 15 meters deep up to a storage tank at ground level. If it pumps 1200 kg of water in 1 minute, what is the mechanical efficiency of the pump system?