### Summary

### 4.1 Displacement and Velocity Vectors

- The position function $\overrightarrow{r}(t)$ gives the position as a function of time of a particle moving in two or three dimensions. Graphically, it is a vector from the origin of a chosen coordinate system to the point where the particle is located at a specific time.
- The displacement vector $\text{\Delta}\overrightarrow{r}$ gives the shortest distance between any two points on the trajectory of a particle in two or three dimensions.
- Instantaneous velocity gives the speed and direction of a particle at a specific time on its trajectory in two or three dimensions, and is a vector in two and three dimensions.
- The velocity vector is tangent to the trajectory of the particle.
- Displacement $\overrightarrow{r}(t)$ can be written as a vector sum of the one-dimensional displacements $\overrightarrow{x}(t),\overrightarrow{y}(t),\overrightarrow{z}(t)$ along the
*x*,*y*, and*z*directions. - Velocity $\overrightarrow{v}(t)$ can be written as a vector sum of the one-dimensional velocities ${v}_{x}(t),{v}_{y}(t),{v}_{z}(t)$ along the
*x*,*y*, and*z*directions. - Motion in any given direction is independent of motion in a perpendicular direction.

### 4.2 Acceleration Vector

- In two and three dimensions, the acceleration vector can have an arbitrary direction and does not necessarily point along a given component of the velocity.
- The instantaneous acceleration is produced by a change in velocity taken over a very short (infinitesimal) time period. Instantaneous acceleration is a vector in two or three dimensions. It is found by taking the derivative of the velocity function with respect to time.
- In three dimensions, acceleration $\overrightarrow{a}(t)$ can be written as a vector sum of the one-dimensional accelerations ${a}_{x}(t),{a}_{y}(t),\text{and}\phantom{\rule{0.2em}{0ex}}{a}_{z}(t)$ along the
*x-*,*y*-, and*z-*axes. - The kinematic equations for constant acceleration can be written as the vector sum of the constant acceleration equations in the
*x*,*y*, and*z*directions.

### 4.3 Projectile Motion

- Projectile motion is the motion of an object subject only to the acceleration of gravity, where the acceleration is constant, as near the surface of Earth.
- To solve projectile motion problems, we analyze the motion of the projectile in the horizontal and vertical directions using the one-dimensional kinematic equations for
*x*and*y*. - The time of flight of a projectile launched with initial vertical velocity ${v}_{0y}$ on an even surface is given by

$${T}_{tof}=\frac{2({v}_{0}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta )}{g}.$$

This equation is valid only when the projectile lands at the same elevation from which it was launched. - The maximum horizontal distance traveled by a projectile is called the range. Again, the equation for range is valid only when the projectile lands at the same elevation from which it was launched.

### 4.4 Uniform Circular Motion

- Uniform circular motion is motion in a circle at constant speed.
- Centripetal acceleration ${\overrightarrow{a}}_{\text{C}}$ is the acceleration a particle must have to follow a circular path. Centripetal acceleration always points toward the center of rotation and has magnitude ${a}_{\text{C}}={v}^{2}\text{/}r.$
- Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. This acceleration is called tangential acceleration ${\overrightarrow{a}}_{\text{T}}.$ The magnitude of tangential acceleration is the time rate of change of the magnitude of the velocity. The tangential acceleration vector is tangential to the circle, whereas the centripetal acceleration vector points radially inward toward the center of the circle. The total acceleration is the vector sum of tangential and centripetal accelerations.
- An object executing uniform circular motion can be described with equations of motion. The position vector of the object is $\overrightarrow{r}(t)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\omega t\widehat{i}+A\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\omega t\widehat{j},$ where
*A*is the magnitude $\left|\overrightarrow{r}(t)\right|,$ which is also the radius of the circle, and $\omega $ is the angular frequency.

### 4.5 Relative Motion in One and Two Dimensions

- When analyzing motion of an object, the reference frame in terms of position, velocity, and acceleration needs to be specified.
- Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies with the choice of reference frame.
- If
*S*and ${S}^{\prime}$ are two reference frames moving relative to each other at a constant velocity, then the velocity of an object relative to*S*is equal to its velocity relative to ${S}^{\prime}$ plus the velocity of ${S}^{\prime}$ relative to*S.* - If two reference frames are moving relative to each other at a constant velocity, then the accelerations of an object as observed in both reference frames are equal.