Energy: Mathematical problems & solutions

Problem 1: Kinetic Energy Calculation

Problem: Calculate the kinetic energy of a car with a mass of $1000\text{kg}$ traveling at a speed of $20\text{m/s}$. Solution: $𝐾𝐸=\frac{1}{2}𝑚{𝑣}^2=\frac{1}{2}\times 1000\text{kg}\times (20\text{m/s}{)}^2=200,000\text{J}$ The kinetic energy of the car is $200,000\text{J}$.

Problem 2: Gravitational Potential Energy

Problem: A book with a mass of $2\text{kg}$ is lifted to a height of $5\text{m}$ above the ground. Calculate its gravitational potential energy. Solution: $𝑃𝐸=𝑚𝑔ℎ=2\text{kg}\times 9.8{\text{m/s}}^2\times 5\text{m}=98\text{J}$ The gravitational potential energy of the book is $98\text{J}$.

Problem 3: Spring Potential Energy

Problem: A spring with a spring constant of $200\text{N/m}$ is compressed by $0.1\text{m}$. Calculate its potential energy. Solution: $𝑃𝐸=\frac{1}{2}𝑘{𝑥}^2=\frac{1}{2}\times 200\text{N/m}\times (0.1\text{m}{)}^2=1\text{J}$ The potential energy of the spring is $1\text{J}$.

Problem 4: Total Mechanical Energy

Problem: A pendulum with a mass of $0.5\text{kg}$ swings from a height of $2\text{m}$. Calculate its total mechanical energy at the highest point. Solution: At the highest point, all of the energy is gravitational potential energy: $𝑃𝐸=𝑚𝑔ℎ=0.5\text{kg}\times 9.8{\text{m/s}}^2\times 2\text{m}=9.8\text{J}$ The total mechanical energy is $9.8\text{J}$.

Problem 5: Conservation of Mechanical Energy

Problem: A roller coaster car with a mass of $500\text{kg}$ starts from rest at a height of $50\text{m}$ on a track with no friction. Calculate its speed at the bottom of the hill. Solution: At the top of the hill, all of the energy is gravitational potential energy: $𝑃𝐸=𝑚𝑔ℎ=500\text{kg}\times 9.8{\text{m/s}}^2\times 50\text{m}=245,000\text{J}$ At the bottom of the hill, all of the potential energy is converted to kinetic energy: $𝐾𝐸=245,000\text{J}$ $\frac{1}{2}𝑚{𝑣}^2=245,000\text{J}$ $𝑣=\sqrt{\frac{2\times 245,000\text{J}}{500\text{kg}}}=\sqrt{490}\approx 22.1\text{m/s}$ The speed at the bottom of the hill is $22.1\text{m/s}$.

Problem 6: Work-Energy Principle

Problem: A force of $50\text{N}$ is applied to push a box $10\text{m}$ across a floor. Calculate the work done. Solution: $𝑊=𝐹𝑑\cos (𝜃)=50\text{N}\times 10\text{m}\times \cos (0^{\circ })=500\text{J}$ The work done is $500\text{J}$.

Problem 7: Power Calculation

Problem: A motor does $5000\text{J}$ of work in $10\text{s}$. Calculate the power. Solution: $𝑃=\frac{𝑊}{𝑡}=\frac{5000\text{J}}{10\text{s}}=500\text{W}$ The power is $500\text{W}$.

Problem 8: Conservation of Energy with Friction

Problem: A block with a mass of $2\text{kg}$ slides down a $30^{\circ }$ incline with a coefficient of kinetic friction of $0.2$. If it starts from rest at a height of $5\text{m}$, calculate its speed at the bottom of the incline. Solution: At the top of the incline, all of the energy is gravitational potential energy: $𝑃𝐸=𝑚𝑔ℎ=2\text{kg}\times 9.8{\text{m/s}}^2\times 5\text{m}=98\text{J}$