Key Points

A scalar quantity has only magnitude, like mass and speed. A vector quantity has both magnitude and direction, like velocity and force, and obeys vector addition laws.

Vectors can be added graphically using the head-to-tail method (triangle law) or by placing their tails together and completing the parallelogram (parallelogram law). Vector addition is commutative: $\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$.

A vector $\mathbf{A}$ can be resolved into components along perpendicular axes. If $\theta$ is the angle with the x-axis, the components are $A_x = A \cos \theta$ and $A_y = A \sin \theta$.

Unit vectors $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ have a magnitude of one and point along the positive x, y, and z-axes, respectively. A vector $\mathbf{A}$ can be written as $\mathbf{A} = A_x\hat{\mathbf{i}} + A_y\hat{\mathbf{j}} + A_z\hat{\mathbf{k}}$.

To add vectors analytically, add their corresponding components. If $\mathbf{R} = \mathbf{A} + \mathbf{B}$, then its components are $R_x = A_x + B_x$ and $R_y = A_y + B_y$.

The magnitude of the resultant of two vectors $\mathbf{A}$ and $\mathbf{B}$ with an angle $\theta$ between them is given by the Law of Cosines: $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$.

The position of a particle is described by a position vector $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}}$. The displacement $\Delta\mathbf{r}$ is the change in position from an initial point to a final point, $\Delta\mathbf{r} = \mathbf{r}' - \mathbf{r}$.

The instantaneous velocity is the time derivative of the position vector, $\mathbf{v} = \frac{d\mathbf{r}}{dt} = v_x\hat{\mathbf{i}} + v_y\hat{\mathbf{j}}$. Its direction is always tangent to the object's path.

The instantaneous acceleration is the time derivative of the velocity vector, $\mathbf{a} = \frac{d\mathbf{v}}{dt} = a_x\hat{\mathbf{i}} + a_y\hat{\mathbf{j}}$. In 2D or 3D motion, the acceleration vector can have any angle relative to the velocity vector.

For constant acceleration $\mathbf{a}$, the velocity and position are given by $\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t$ and $\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0t + \frac{1}{2}\mathbf{a}t^2$.

Projectile motion is the motion of an object thrown into the air, subject only to gravity. It is treated as two independent motions: constant velocity horizontally ($a_x = 0$) and constant downward acceleration vertically ($a_y = -g$).

The trajectory of a projectile is a parabola. The equation of its path is given by $y = (\tan \theta_0)x - \frac{g}{2(v_0 \cos \theta_0)^2}x^2$.

The maximum vertical height $h_m$ reached by a projectile is given by the formula $h_m = \frac{(v_0 \sin \theta_0)^2}{2g}$.

The total time the projectile is in the air before returning to the same vertical level is called the time of flight, given by $T_f = \frac{2v_0 \sin \theta_0}{g}$.

The horizontal distance traveled by the projectile is the range, $R = \frac{v_0^2 \sin(2\theta_0)}{g}$. The range is maximum for a launch angle of $45^{\circ}$.

Uniform circular motion describes an object moving in a circular path at a constant speed. The velocity is not constant because its direction is continuously changing.

An object in uniform circular motion experiences centripetal acceleration, which is always directed towards the center of the circle. Its magnitude is $a_c = \frac{v^2}{R} = \omega^2 R$, where $v$ is linear speed, $R$ is radius, and $\omega$ is angular speed.