# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - Rectilinear motion (motion along a straight line) - Projectile motion under gravity - Relationships between position $s(t)$, velocity $v(t)$, and acceleration $a(t)$ - Differentiation of position to find velocity and acceleration - Integration of acceleration to find velocity and position - Finding when a particle is at rest ($v=0$) - Determining maximum height or extreme positions - Calculating displacement vs. total distance traveled - Constant acceleration kinematic equations (SUVAT equations) --- # **Important Formulas** Below are the key formulas used in this exercise: **Kinematics Relationships:** $$v = \frac{ds}{dt} \quad \text{(Velocity is the derivative of position)}$$ $$a = \frac{dv}{dt} = \frac{d^2s}{dt^2} \quad \text{(Acceleration is the derivative of velocity)}$$ **Integration Formulas:** $$v = \int a \, dt \quad \text{(Velocity from acceleration)}$$ $$s = \int v \, dt \quad \text{(Position from velocity)}$$ **Distance and Displacement:** $$\text{Displacement} = \int_{t_1}^{t_2} v(t) \, dt$$ $$\text{Total Distance} = \int_{t_1}^{t_2} |v(t)| \, dt$$ **Constant Acceleration Equations:** $$v = u + at$$ $$s = ut + \frac{1}{2}at^2$$ $$v^2 = u^2 + 2as$$ **Projectile Motion:** $$h(t) = h_0 + v_0t - \frac{1}{2}gt^2 \quad \text{(where $g \approx 32$ ft/s$^2$ or $9.8$ m/s$^2$)}$$ --- # **Summary** This exercise covers the application of calculus to kinematics problems involving particles moving along a straight line and projectile motion. The key strategy involves understanding the fundamental relationships: differentiation connects position $\to$ velocity $\to$ acceleration, while integration reverses this process. Common problem types include finding when objects are at rest (solve $v=0$), determining maximum height or position (find critical points of $s(t)$ or when $v=0$), calculating total distance traveled (integrate $|v|$ between direction changes), and distinguishing between displacement and total distance. For constant acceleration problems, standard kinematic equations apply, while variable acceleration requires integration techniques.