Problem Analysis:
The given equations are related to the kinematic equations of motion. The task involves solving for the time interval \( \Delta t \) using the provided equations. Let's break it down step by step.
#### Given Equations:
1. \( \Delta y = v_i \Delta t + \frac{1}{2} a \Delta t^2 \)
2. \( \Delta y = \frac{1}{2} a \Delta t^2 \) (This is derived by setting \( v_i = 0 \))
#### Objective:
Solve for \( \Delta t \) in terms of \( \Delta y \) and \( a \).
---
Step-by-Step Solution:
#### Step 1: Start with the simplified equation
From the second equation:
\[
\Delta y = \frac{1}{2} a \Delta t^2
\]
#### Step 2: Isolate \( \Delta t^2 \)
To isolate \( \Delta t^2 \), multiply both sides of the equation by 2:
\[
2 \Delta y = a \Delta t^2
\]
Next, divide both sides by \( a \):
\[
\frac{2 \Delta y}{a} = \Delta t^2
\]
#### Step 3: Solve for \( \Delta t \)
Take the square root of both sides to solve for \( \Delta t \):
\[
\Delta t = \sqrt{\frac{2 \Delta y}{a}}
\]
---
Final Answer:
\[
\boxed{\Delta t = \sqrt{\frac{2 \Delta y}{a}}}
\]
---
Explanation:
1.
Initial Equation: The first equation is the general kinematic equation for displacement under constant acceleration:
\[
\Delta y = v_i \Delta t + \frac{1}{2} a \Delta t^2
\]
When the initial velocity \( v_i \) is zero, the term \( v_i \Delta t \) vanishes, simplifying the equation to:
\[
\Delta y = \frac{1}{2} a \Delta t^2
\]
2.
Simplification: By setting \( v_i = 0 \), we directly use the simplified form to solve for \( \Delta t \).
3.
Algebraic Manipulation: Solving for \( \Delta t \) involves isolating \( \Delta t^2 \) and then taking the square root. This process ensures that the solution is consistent with the physical scenario where the object starts from rest.
4.
Physical Interpretation: The final expression \( \Delta t = \sqrt{\frac{2 \Delta y}{a}} \) gives the time required for an object starting from rest to travel a distance \( \Delta y \) under constant acceleration \( a \).
Thus, the solution is both mathematically correct and physically meaningful.
Parent Tip: Review the logic above to help your child master the concept of math problem algebra 2 equations.
