Problem 20 - Applied Physics

Question

Solve each formula for the quantity given. $$ v_{2}=v_{1}+a t \text { for } t $$

Step-by-Step Solution

Answer

\( t = \frac{v_{2} - v_{1}}{a} \)

Step 1: Identify the formula

The formula given is \( v_{2} = v_{1} + a t \). We need to solve this formula for the variable \( t \). This means we need to express \( t \) in terms of the other variables \( v_{2}, v_{1}, \) and \( a \).

Step 2: Isolate the term containing \( t \)

To solve for \( t \), we first need to isolate the term \( a t \) on one side of the equation. We can do this by subtracting \( v_{1} \) from both sides of the equation:\[ v_{2} - v_{1} = a t \]

Step 3: Solve for \( t \)

Now that we have the equation \( v_{2} - v_{1} = a t \), we need to solve for \( t \) by dividing both sides by \( a \) (assuming \( a eq 0 \)):\[ t = \frac{v_{2} - v_{1}}{a} \]

Key Concepts

Kinematic Equations

Kinematic equations describe the motion of objects using variables such as velocity, acceleration, time, and displacement. They are crucial in understanding how physical bodies move under constant acceleration. In the exercise, the equation \( v_{2} = v_{1} + at \) is a typical kinematic equation where:

• \( v_{1} \) is the initial velocity,
• \( v_{2} \) is the final velocity,
• \( a \) is the acceleration,
• \( t \) is the time.

By understanding these variables and how they relate to one another, we can model and predict the behaviors of moving objects. Kinematic equations like this one allow us to solve for unknown variables, given the others, when motion is uniform.

Solving for Variables

In physics, solving for a specific variable within an equation is a common task. It involves rearranging the equation so the desired variable is by itself on one side of the equation. This exercise involves solving for the variable \( t \) in the context of the kinematic equation \( v_{2} = v_{1} + at \).To do this, follow these steps:

• Identify the variable you need to solve for—in this case, \( t \).
• Isolate the term of the equation that contains \( t \). In our example, subtract \( v_{1} \) from both sides to have \( at \) alone.
• Finish isolating \( t \) by dividing both sides by \( a \), resulting in \( t = \frac{v_{2} - v_{1}}{a} \).

These steps help you systematically solve for a variable while ensuring algebraic accuracy.

Algebraic Manipulation

Algebraic manipulation is the technique used to rearrange equations to make one variable the subject. It consists of a series of operations that include adding, subtracting, multiplying, dividing, and moving terms across the equation to isolate the desired variable.Here, we performed algebraic manipulation to solve for \( t \):

• First, subtract \( v_{1} \) from both sides to handle the terms not related to \( t \).
• Next, divide by \( a \) to isolate \( t \) since \( a \) applies to \( t \) through multiplication.

Mastering algebraic manipulation helps simplify complex equations and makes the process of solving physics problems more straightforward.

Acceleration and Velocity Relationships

Acceleration and velocity are fundamental concepts in physics that describe the change in motion of objects. In the kinematic equation \( v_{2} = v_{1} + at \), we see:- **Velocity** \((v)\): Measures how fast an object is moving and in what direction. Initial velocity \( v_{1} \) is the starting speed, and final velocity \( v_{2} \) is the speed after some time.- **Acceleration** \((a)\): The rate of change of velocity. Constant acceleration means the velocity changes at a consistent rate over time \( t \).Understanding these relationships allows us to determine how an object reaches certain speeds or how long it takes to reach those speeds. Acceleration can affect travel time and distance, which is why it's useful to calculate \( t \) using these variables.