Elastic Energy, Velocity, And Height Calculations In Physics

Hey guys! Let's dive into some physics problems involving elastic energy, velocity, and height. We'll break down each problem step-by-step, making sure you understand the concepts and how to apply the formulas. Get ready to put on your thinking caps!

Calculating the Spring Constant

Spring constant is a measure of how stiff a spring is. When you compress or stretch a spring, it stores potential energy, known as elastic energy. The formula to calculate elastic energy (${E}$) is:

${ E = \frac{1}{2} k x^2 }$

Where:

• ${E}$ is the elastic energy stored in the spring (in Joules).
• ${k}$ is the spring constant (in N/m).
• ${x}$ is the displacement or compression of the spring from its equilibrium position (in meters).

Now, let's apply this to our problem:

Problem: A spring is compressed to 2 meters. Knowing that its elastic energy is 1200J, what is the spring constant?

Solution:

We are given:

• ${E = 1200 \text{ J}}$
• ${x = 2 \text{ m}}$

We need to find ${k}$.

Rearrange the formula to solve for ${k}$:

${ k = \frac{2E}{x^2} }$

Plug in the values:

${ k = \frac{2 \times 1200}{2^2} = \frac{2400}{4} = 600 \text{ N/m} }$

So, the spring constant of the spring is 600 N/m. This means that it takes 600 Newtons of force to compress (or stretch) the spring by 1 meter. Understanding spring constants is super important in various applications, from designing suspension systems in cars to understanding how trampolines work. Remember, the higher the spring constant, the stiffer the spring, and the more force it takes to compress or stretch it. Elastic energy is stored when the spring is deformed and is released when the spring returns to its original shape, making it a fundamental concept in mechanics.

Determining the Velocity of a Panda

Alright, let's switch gears and talk about kinetic energy and velocity. Kinetic energy is the energy an object possesses due to its motion. The formula for kinetic energy (${KE}$) is:

${ KE = \frac{1}{2} m v^2 }$

Where:

• ${KE}$ is the kinetic energy (in Joules).
• ${m}$ is the mass of the object (in kg).
• ${v}$ is the velocity of the object (in m/s).

Let's tackle this problem:

Problem: Calculate the velocity of a lame panda that runs with 9000J of energy and has a mass of 100 kg.

Solution:

We know:

• ${KE = 9000 \text{ J}}$
• ${m = 100 \text{ kg}}$

We need to find ${v}$.

Rearrange the formula to solve for ${v}$:

${ v = \sqrt{\frac{2KE}{m}} }$

Plug in the values:

${ v = \sqrt{\frac{2 \times 9000}{100}} = \sqrt{\frac{18000}{100}} = \sqrt{180} \approx 13.42 \text{ m/s} }$

Therefore, the velocity of the lame panda is approximately 13.42 m/s. Keep in mind that kinetic energy is directly related to both the mass and the square of the velocity. This means that if you double the mass, you double the kinetic energy, but if you double the velocity, you quadruple the kinetic energy! It’s all about how fast and how heavy something is when it's moving. Understanding these concepts is crucial in fields like transportation, sports science, and even in understanding the motion of celestial bodies. So next time you see something moving, remember it's all about kinetic energy!

Calculating Height Using Potential Energy

Now, let's explore potential energy, specifically gravitational potential energy. Gravitational potential energy (${PE}$) is the energy an object has due to its position above the ground. The formula for gravitational potential energy is:

${ PE = mgh }$

Where:

• ${PE}$ is the potential energy (in Joules).
• ${m}$ is the mass of the object (in kg).
• ${g}$ is the acceleration due to gravity (approximately 9.8 m/s²).
• ${h}$ is the height of the object above the ground (in meters).

Imagine lifting a heavy box; the higher you lift it, the more potential energy it has. When you release the box, this potential energy converts into kinetic energy as it falls. Let’s consider a problem to calculate height:

Problem: Suppose an object with a mass of ${ m }$ has a potential energy of ${ PE }$. What is the height ${ h }$ of the object?

Solution:

We know:

• ${PE}$ = Gravitational Potential Energy (in Joules)
• ${m}$ = Mass of the object (in kg)
• ${g}$ = Acceleration due to gravity (${ \approx 9.8 \text{ m/s}^2 }$)

We need to find ${h}$.

Rearrange the formula to solve for ${h}$:

${ h = \frac{PE}{mg} }$

Let's assume we have an object with a mass of 50 kg and a potential energy of 4900 J.

Plug in the values:

${ h = \frac{4900}{50 \times 9.8} = \frac{4900}{490} = 10 \text{ m} }$

So, the height of the object is 10 meters. Understanding gravitational potential energy helps us grasp concepts like how dams store energy (water at a height has potential energy) and how roller coasters work (converting potential energy at the top of a hill into kinetic energy as they descend). The relationship between potential energy, mass, gravity, and height is fundamental in understanding various real-world scenarios. Always remember, potential energy is all about position, and it can be converted into other forms of energy, making it a crucial concept in physics.

Conclusion

So there you have it, guys! We've tackled problems involving elastic energy, kinetic energy, and gravitational potential energy. Remember these formulas and concepts, and you'll be well on your way to mastering physics. Keep practicing, and don't be afraid to ask questions. Physics is all about understanding the world around us, so keep exploring!