Study guide: Rectilinear and Curvilinear Motion

📋 Course Outline

1. Rectilinear Motion Fundamentals
2. Position and Displacement
3. Velocity and Speed
4. Acceleration and Motion Equations
5. Integration and Constant Acceleration
6. Free Fall
7. Curvilinear Motion Coordinates
8. Projectile Motion

📖 1. Rectilinear Motion Fundamentals

🔑 Key Concepts & Definitions

• Rectilinear motion: motion along a straight line in one dimension, described using one axis and one position parameter
• Particle: an infinitely small object that has mass and can undergo translation but not rotation

📖 2. Position and Displacement

🔑 Key Concepts & Definitions

• Position vector: the vector from the origin of the reference frame to that point, and in rectilinear motion it corresponds to the distance from the reference point

📝 Essential Points

📐 Formula - The displacement vector between times $t_1$ and $t_2$ is $\Delta\vec r=\vec r(t_2)-\vec r(t_1)$, with $\Delta x=x(t_2)-x(t_1)$, $\Delta y=y(t_2)-y(t_1)$, and $\Delta z=z(t_2)-z(t_1)$.

💡 Memory Hook

Position locates the particle; displacement compares two positions.

📖 3. Velocity and Speed

📝 Essential Points

📐 Formula - The average velocity is $\vec v_{avg}=\frac{\Delta\vec r}{\Delta t}$, and in one dimension its component is $v_{avg}=\frac{\Delta x}{\Delta t}$.

📐 Formula - The instantaneous velocity is $\vec v(t)=\frac{d\vec r}{dt}$, and in one dimension it is $v(t)=\frac{dx}{dt}$; it is the slope of the position–time curve.

📌 Average speed is total distance travelled divided by total time and is always positive, whereas average velocity is displacement divided by elapsed time and can be positive or negative.

💡 Memory Hook

Velocity can be positive or negative; speed is always positive.

📖 4. Acceleration and Motion Equations

📝 Essential Points

📐 Formula - The instantaneous acceleration is $\vec a(t)=\frac{d\vec v}{dt}=\frac{d^2\vec r}{dt^2}$, and in one dimension it is $a(t)=\frac{d^2x}{dt^2}$.

📌 Negative acceleration means that a particle moving in the positive direction is slowing down or that a particle moving in the negative direction is speeding up.

💡 Memory Hook

Acceleration changes velocity, and velocity changes position.

📖 5. Integration and Constant Acceleration

★ Must-know

📐 Formula - For constant acceleration, the equations of motion are $v=v_0+at$, $x=x_0+v_0t+\frac{1}{2}at^2$, and $v^2=v_0^2+2a(x-x_0)$.

• Integrating acceleration over time gives the change in velocity, because the area under an acceleration–time curve equals the velocity change.

Further detail

📐 Formula - For constant acceleration, the velocity change is $v(t_B)-v(t_A)=\int_{t_A}^{t_B}a(t)\,dt$.

💡 Memory Hook

Acceleration → velocity → position

📖 6. Free Fall

★ Must-know

📐 Formula - For vertical free fall with the positive direction upward, the equations are $v=v_0-gt$, $y=y_0+v_0t-\frac{1}{2}gt^2$, and $v^2=v_0^2-2g(y-y_0)$.

• Near the Earth’s surface, all objects have the same gravitational acceleration in the absence of air resistance, with $g=9.81\,\mathrm{m\,s^{-2}}$.

Further detail

📐 Formula - Newton’s gravitational acceleration at the Earth’s surface is $g=\frac{GM_E}{R^2}=9.81\,\mathrm{m\,s^{-2}}$, with $G=6.67\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$. - law of universal gravitation

💡 Memory Hook

Air resistance changes fall times; gravity gives all objects the same acceleration near Earth.

📖 7. Curvilinear Motion Coordinates

★ Must-know

📐 Formula - In Cartesian coordinates, the position, velocity, and acceleration vectors are $\vec r=x\hat i+y\hat j+z\hat k$, $\vec v=\frac{d\vec r}{dt}$, and $\vec a=\frac{d^2\vec r}{dt^2}$.

📌 Instantaneous velocity is tangent to the trajectory, while instantaneous speed is the positive magnitude of the velocity vector, $|\vec v|=\sqrt{v_x^2+v_y^2+v_z^2}$.

Further detail

📐 Formula - The average acceleration is $\vec a_{avg}=\frac{\Delta\vec v}{\Delta t}$, while instantaneous acceleration is $\vec a=\frac{d\vec v}{dt}=\frac{d^2\vec r}{dt^2}$.

💡 Memory Hook

Position → velocity → acceleration

📖 8. Projectile Motion

🔑 Key Concepts & Definitions

• Projectile motion: the motion of a particle launched with an initial velocity while its acceleration remains the free-fall acceleration $\vec g=-g\hat j$

📝 Essential Points

📐 Formula - For projectile motion, the velocity components are $v_x(t)=v_0\cos\theta$ and $v_y(t)=v_0\sin\theta-gt$, while the position components are $x(t)=x_0+v_0\cos\theta\,t$ and $y(t)=y_0+v_0\sin\theta\,t-\frac{1}{2}gt^2$.

📐 Formula - When the projectile starts and lands at the same height, its flight time is $t_C=\frac{2v_0\sin\theta}{g}$, its maximum range is $x_{C,max}=\frac{v_0^2}{g}$ at $\theta=45^\circ$, and its maximum height is $h_{max}=\frac{v_0^2\sin^2\theta}{2g}$.

• Projectile motion can be solved as two independent one-dimensional motions: horizontal motion with zero acceleration and vertical motion with acceleration $-g$.

💡 Memory Hook

Horizontal motion is uniform; vertical motion is accelerated by gravity.

📊 Synthesis Tables

Velocity and speed comparison

Quantity	Definition	Sign or direction
Average velocity	Displacement divided by elapsed time	Can be positive or negative and is a vector
Average speed	Total distance divided by elapsed time	Always positive
Instantaneous velocity	Derivative of position	Tangent vector to the trajectory
Instantaneous speed	Magnitude of instantaneous velocity	Positive scalar