1. **Stating the problem:** We are given two kinematic equations to analyze and understand their use in motion problems. 2. **Equations provided:** - Displacement: $$x = V_i t + \frac{1}{2} a t^2$$ - Velocity as a function of displacement: $$V_f^2 = V_i^2 + 2 a x$$ 3. **Explanation and formulas:** - The first equation calculates displacement $$x$$ when initial velocity $$V_i$$, acceleration $$a$$, and time $$t$$ are known. - The second equation relates final velocity $$V_f$$ to initial velocity $$V_i$$, acceleration $$a$$, and displacement $$x$$ without involving time. 4. **Important rules:** - Acceleration $$a$$ is constant. - Initial velocity $$V_i$$ and final velocity $$V_f$$ are velocities at the start and end of the time interval. - Time $$t$$ is the duration of motion. 5. **Intermediate work example:** Suppose $$V_i=5$$ m/s, $$a=2$$ m/s², and $$t=3$$ s. - Calculate displacement: $$x = 5 \times 3 + \frac{1}{2} \times 2 \times 3^2 = 15 + \frac{1}{2} \times 2 \times 9 = 15 + 9 = 24$$ m - Calculate final velocity using displacement: $$V_f^2 = 5^2 + 2 \times 2 \times 24 = 25 + 96 = 121$$ $$V_f = \sqrt{121} = 11$$ m/s 6. **Summary:** - Use the first formula to find displacement when time is known. - Use the second formula to find final velocity when displacement is known. Final answers for the example: displacement $$x=24$$ m, final velocity $$V_f=11$$ m/s.