Problem 9 - Applied Physics

Question

Solve each formula for the quantity given. $$ v^{2}=2 g h \text { for } h $$

Step-by-Step Solution

Answer

The formula solved for \(h\) is \( h = \frac{v^2}{2g} \).

Step 1: Identify the formula

We are given the formula \(v^2 = 2gh\). Our task is to solve it for \(h\). This means we need to rearrange the formula so that \(h\) is isolated on one side of the equation.

Step 2: Divide both sides by the coefficient of \(h\)

In the equation \(v^2 = 2gh\), the term \(2g\) is the coefficient of \(h\). Divide both sides of the equation by this coefficient to isolate \(h\). This gives us:\[h = \frac{v^2}{2g}\]

Key Concepts

Solve for a Variable

Solving for a variable means finding the value of a specific variable in an equation. This variable is often surrounded by other numbers or variables.
Our goal is to isolate the variable on one side of the equation. This makes it easier to see what the variable equals.
For example, if we have the equation \(x + 3 = 5\), we can solve for \(x\) by subtracting 3 from both sides. This gives us \(x = 2\).
To successfully solve for a variable, follow these steps:

• Identify the variable you need to solve for.
• Perform operations that will leave the variable by itself on one side of the equation.
• Simplify the equation as needed.

These steps help us understand what the variable should be in simple and clear terms.

Rearranging Equations

Rearranging equations is a method used to make an equation easier to understand, by moving terms around. This often involves changing the position of numbers or variables across the equals sign.
It is important to always perform the same mathematical operation on both sides of the equation. This keeps the equation balanced, ensuring that it remains true.
For instance, consider the equation \(v^2 = 2gh\). If we want to solve for \(h\), we need to rearrange it by dividing both sides by \(2g\). This rearrangement gives us \(h = \frac{v^2}{2g}\).
Key tips for rearranging equations include:

• Identify the term you want to move.
• Perform inverse operations to isolate variables.
• Ensure consistency by applying changes to both sides.

By rearranging equations, we can gain insights into the relationships between different components within the formula.

Kinematics Equations

Kinematics equations are tools in physics used to describe the motion of objects. These equations often involve variables like distance, velocity, time, and acceleration.
They help us predict the future position or velocity of an object when its initial state and motion conditions are known.
A common kinematics equation used is \(v^2 = u^2 + 2as\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(s\) is the displacement.
Understanding kinematics is essential for analyzing how objects move in various environments, be it on earth or in outer space. These equations:

• Allow us to calculate unknown variables.
• Help predict motion outcomes.
• Are crucial in engineering and scientific applications.

With these equations, we can explore the dynamics of moving particles, improving our understanding of the world around us.

Gravitational Acceleration

Gravitational acceleration is a specific type of acceleration that occurs due to gravity. On Earth, this acceleration usually has a constant value of approximately \(9.8 \text{ m/s}^2\).
This means that in the absence of air resistance, an object in free fall accelerates downwards at this rate.
In physics, gravitational acceleration is often represented by the symbol \(g\) and appears in many equations involving falling bodies, such as \(v^2 = 2gh\).
Key points about gravitational acceleration include:

• It acts downwards towards the center of the Earth.
• It's a constant value near the Earth's surface.
• Significantly affects projectile motion and free-fall dynamics.

Understanding gravitational acceleration helps us calculate how fast objects increase their speed as they fall, essential for many scientific and engineering calculations.