1. **Stating the problem:** We want to find the velocity and height of an object in motion, typically under gravity. 2. **Formulas used:** - Velocity at time $t$: $$v(t) = v_0 - gt$$ where $v_0$ is the initial velocity and $g$ is the acceleration due to gravity (approximately 9.8 m/s²). - Height at time $t$: $$h(t) = h_0 + v_0 t - \frac{1}{2}gt^2$$ where $h_0$ is the initial height. 3. **Explanation:** - Velocity changes over time due to gravity pulling the object down. - Height changes based on initial height, initial velocity, and the effect of gravity. 4. **Intermediate work example:** Suppose $v_0 = 20$, $h_0 = 0$, and $t = 2$ seconds. - Calculate velocity: $$v(2) = 20 - 9.8 \times 2 = 20 - 19.6 = 0.4$$ - Calculate height: $$h(2) = 0 + 20 \times 2 - \frac{1}{2} \times 9.8 \times 2^2 = 40 - \frac{1}{2} \times 9.8 \times 4 = 40 - 19.6 = 20.4$$ 5. **Summary:** - Use $v(t) = v_0 - gt$ to find velocity at any time. - Use $h(t) = h_0 + v_0 t - \frac{1}{2}gt^2$ to find height at any time. - Plug in known values and simplify step-by-step.