Question

List some formulas that occur in applications. Solve each formula for the indicated variable. $$ \text { Mechanics } v=-g t+v_{0} \quad \text { for } t $$

Short answer

The formula for \( t \) is \( t = \frac{v_0 - v}{g} \).

Step-by-step solution

Write down the given formula

The given formula is from mechanics: \[ v = -g t + v_0 \]We need to solve for the variable \( t \).

Rearrange the equation

First, isolate the term with \( t \). Subtract \( v_0 \) from both sides of the equation: \[ v - v_0 = -g t \]

Solve for \( t \)

Next, divide both sides by \( -g \) to solve for \( t \): \[ t = \frac{v - v_0}{-g} \]or simplified: \[ t = \frac{v_0 - v}{g} \]

Key concepts

mechanics formulas

Mechanics formulas are used to describe the motion of objects. They help us understand how forces, velocities, and accelerations interact.

• These formulas can predict how an object might move over time.
• They also relate different physical quantities such as time, distance, velocity, and acceleration.

In the given formula, \( v = -g t + v_0 \), it describes the velocity of an object under the influence of gravity. Here, \( v_0 \) represents the initial velocity, \( -g \) is the acceleration due to gravity (a negative value), and \( t \) is the time.

variable isolation

Variable isolation is the process of solving for a specific variable in an equation. It involves moving all other terms to the opposite side of the equation to leave the desired variable by itself.
In the given problem, we need to isolate \( t \) in the formula \( v = -g t + v_0 \).

• First, isolate the term with \( t \) by subtracting \( v_0 \) from both sides, giving: \( v - v_0 = -g t \).
• Next, divide both sides by \( -g \) to solve for \( t \): \[ t = \frac{v - v_0}{-g} \].

Variable isolation is crucial in solving scientific and engineering problems by focusing on one parameter at a time.

rearranging equations

Rearranging equations is a technique used to organize an equation to a more desired form.
This is often applied when solving for a particular variable, making complex problems more manageable.

• This step involves adding, subtracting, multiplying, or dividing both sides of the equation.
• In the example \( v = -g t + v_0 \), we rearrange by isolating \( t \) first through subtraction and then division: \[ t = \frac{v - v_0}{-g} \].

Such manipulations ensure that we get the necessary variable in terms that are easy to understand and use in further calculations.

kinematic equations

Kinematic equations describe the motion of objects without considering the causes of the motion.
They primarily focus on the relationships between displacement, velocity, acceleration, and time.

• The formula \( v = -g t + v_0 \) is one of many kinematic equations.
• These equations provide a critical understanding of how objects move, helping to predict future positions and velocities based on known values.
  This exploration helps in various fields such as physics, engineering, and even sports science.

Becoming familiar with kinematic equations is fundamental for solving real-world physics problems.