Question

37 sep obtain and evaluate information often, a third equation of motion is used when solving constant acceleration problems: ( v^2 = v_1^2 + 2adelta d ). go online to find how the equation is derived. does it come directly from analyzing the motion graphs, or is it derived from the other two equations of motion? when would it be useful?

Explanation:

1. Derivation Source: The equation \( v^2 = v_i^2 + 2a\Delta d \) is derived from the other two kinematic equations (\( v = v_i + at \) and \( \Delta d=v_i t+\frac{1}{2}at^2 \)), not directly from motion graphs. Solve \( v = v_i + at \) for \( t=\frac{v - v_i}{a} \), substitute into \( \Delta d = v_i t+\frac{1}{2}at^2 \), and simplify to get \( v^2 = v_i^2 + 2a\Delta d \).
2. Usefulness: It’s useful when time (\( t \)) is unknown, as it relates initial velocity (\( v_i \)), final velocity (\( v \)), acceleration (\( a \)), and displacement (\( \Delta d \)) without needing to calculate or know time. For example, in problems like finding the final velocity of a car accelerating over a known distance (without knowing the time taken) or the stopping distance of a vehicle (where time to stop isn’t needed).

Answer:

• Derivation: Derived from the other two kinematic equations (\( v = v_i + at \) and \( \Delta d = v_i t+\frac{1}{2}at^2 \)), not directly from motion graphs.
• Usefulness: Useful when time (\( t \)) is unknown, relating \( v_i \), \( v \), \( a \), and \( \Delta d \) (e.g., finding final velocity over a known distance or stopping distance).