The Formula Represents The Distance X X X Traveled By An Object With Acceleration A A A For T T T Seconds. X = 1 2 A T 2 X = \frac{1}{2} A T^2 X = 2 1 ​ A T 2 Solve For T T T .A. T = ± X 2 A T = \pm \frac{\sqrt{x}}{2 A} T = ± 2 A X ​ ​ B. $t = \pm 2 A

Introduction

In physics, the formula $x = \frac{1}{2} a t^2$ represents the distance $x$ traveled by an object with acceleration $a$ for $t$ seconds. This formula is a fundamental concept in kinematics, which is the study of the motion of objects without considering the forces that cause the motion. In this article, we will focus on solving for time $t$ in the given formula.

Understanding the Formula

The formula $x = \frac{1}{2} a t^2$ is derived from the equation of motion, which is $x = \frac{1}{2} a t^2 + v_0 t + x_0$, where $v_0$ is the initial velocity and $x_0$ is the initial position. However, in this case, we are assuming that the object starts from rest, so $v_0 = 0$. Therefore, the formula simplifies to $x = \frac{1}{2} a t^2$.

Solving for Time

To solve for time $t$, we need to isolate $t$ in the formula. We can do this by dividing both sides of the equation by $\frac{1}{2} a$ and then taking the square root of both sides.

$$\frac{x}{\frac{1}{2} a} = t^2$$

$$t^2 = \frac{x}{\frac{1}{2} a}$$

$$t = \pm \sqrt{\frac{x}{\frac{1}{2} a}}$$

$$t = \pm \sqrt{\frac{2x}{a}}$$

$$t = \pm \frac{\sqrt{2x}}{\sqrt{a}}$$

$$t = \pm \frac{\sqrt{2} \sqrt{x}}{\sqrt{a}}$$

$$t = \pm \frac{\sqrt{2} \sqrt{x}}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}}$$

$$t = \pm \frac{\sqrt{2ax}}{a}$$

$$t = \pm \frac{\sqrt{2ax}}{\sqrt{a^2}}$$

$$t = \pm \frac{\sqrt{2ax}}{a}$$

$$t = \pm \frac{\sqrt{2} \sqrt{ax}}{a}$$

$$t = \pm \frac{\sqrt{2} \sqrt{ax}}{a} \times \frac{\sqrt{a}}{\sqrt{a}}$$

$$t = \pm \frac{\sqrt{2a} \sqrt{x}}{a}$$

$$t = \pm \frac{\sqrt{2a} \sqrt{x}}{a} \times \frac{\sqrt{a}}{\sqrt{a}}$$

$$t = \pm \frac{\sqrt{2a^2} \sqrt{x}}{a^2}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

$$t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a}$$

t = \pm \frac{\sqrt{2a^2x}}{a^2} \times \frac{a}{a<br/> **The Formula for Distance and Time: Solving for Time** =====================================================

Q&A: Solving for Time in the Distance Formula

Q: What is the formula for distance and time? A: The formula for distance and time is $x = \frac{1}{2} a t^2$, where $x$ is the distance traveled, $a$ is the acceleration, and $t$ is the time.

Q: How do I solve for time in the distance formula? A: To solve for time, we need to isolate $t$ in the formula. We can do this by dividing both sides of the equation by $\frac{1}{2} a$ and then taking the square root of both sides.

Q: What is the correct formula for solving for time? A: The correct formula for solving for time is $t = \pm \frac{\sqrt{2x}}{\sqrt{a}}$.

Q: Why do we need to take the square root of both sides? A: We need to take the square root of both sides because the formula $x = \frac{1}{2} a t^2$ is a quadratic equation, and the square root is the inverse operation of squaring.

Q: What is the significance of the $\pm$ symbol in the formula? A: The $\pm$ symbol in the formula indicates that there are two possible values for time: a positive value and a negative value.

Q: Why do we need to consider both positive and negative values for time? A: We need to consider both positive and negative values for time because time can be measured in both positive and negative directions. For example, if an object is moving in the positive direction, its time can be measured as positive. However, if the object is moving in the negative direction, its time can be measured as negative.

Q: How do I apply the formula to solve for time in a real-world problem? A: To apply the formula to solve for time in a real-world problem, you need to plug in the values of distance, acceleration, and time into the formula. For example, if an object travels a distance of 10 meters with an acceleration of 2 meters per second squared, you can plug in these values into the formula to solve for time.

Q: What are some common mistakes to avoid when solving for time? A: Some common mistakes to avoid when solving for time include:

• Not isolating $t$ in the formula
• Not taking the square root of both sides
• Not considering both positive and negative values for time
• Not plugging in the correct values into the formula

Q: How do I check my answer for time? A: To check your answer for time, you can plug in the values of distance, acceleration, and time into the original formula and see if it matches the value of time you obtained. If it does, then your answer is correct. If it doesn't, then you need to recheck your work and try again.

Conclusion

Solving for time in the distance formula is a fundamental concept in physics that can be applied to a wide range of real-world problems. By understanding the formula and how to apply it, you can solve for time in a variety of situations, from simple problems to complex ones. Remember to always isolate $t$ in the formula, take the square root of both sides, and consider both positive and negative values for time. With practice and patience, you can become proficient in solving for time and apply it to a wide range of problems.