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: $v=\frac{ds}{dt}\text{(Velocity is the derivative of position)}$ $a=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}\text{(Acceleration is the derivative of velocity)}$

Integration Formulas: $v=\int adt\text{(Velocity from acceleration)}$

$s=\int vdt\text{(Position from velocity)}$

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: $v=u+at$ $s=ut+\frac{1}{2}a{t}^2$ $v^2=u^2+2as$

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.