Answer to Question #162857 in Mechanics | Relativity for jireh

Question

cite 5 examples of motion in 2 to 3 dimensions. illustrate your 5 examples to position displacement and acceleration in vector notation

Answer 1

The position of a particle can be described by a position vector, with respect to a reference origin. 

$\overrightarrow{r} = x\hat{i}+ y\hat{j}+ z\hat{k}$

The displacement of a particle is the change of the position vector during a certain time.

$\Delta\overrightarrow{r} = (x_2\hat{i}+ y_2\hat{j}+ z_2\hat{k})- (x_1\hat{i}+ y_1\hat{j}+ z_1\hat{k})$

$\Delta\overrightarrow{r} = (x_2-x_1)\hat{i}+ (y_2-y_1)\hat{j}+ (z_2-z_1)\hat{k}$

Following the same definition as in average velocity, 

$average\space acceleration = \large\frac{change\space in \space velocity}{time \space interval}$

$\overrightarrow{a}_{avg} = \large\frac{\overrightarrow{v}_2 -\overrightarrow{v}_1 }{\Delta t} = \frac{\Delta\overrightarrow{v} }{\Delta t}$

If we shrink Δt to zero, then the average acceleration value approaches to the instant acceleration value, which is the derivative of velocity with respect to time: 

$\overrightarrow{a} = \large\frac{d\overrightarrow{v} }{dt}$

$\overrightarrow{a} = \large\frac{d}{dt}$ $(v_x\hat{i}+v_y\hat{j}+v_z\hat{k})$ =

$= \large(\frac{dv_x}{dt}\hat{i}+\frac{dv_y}{dt}\hat{j}+\frac{dv_z}{dt}\hat{k})$

$= (a_x\hat{i}+a_y\hat{j}+a_z\hat{k})$

examples of motion in 2 dimension: Circular Motion, Projectile Motion, Uniform Circular Motion

In Projectile motion:

Examples in sports: Tennis, Baseball, Football, Lacrosse, Racquetball, Soccer…

A particle moves in a vertical plane, with some initial velocity 𝒗𝟎; the only acceleration is the free-fall acceleration, 𝒈, directed vertically downward. 

The trajectory of a projectile motion is a parabola unless the object has no horizontal component of motion in which case it is simply free fall.

The equation for the trajectory: $y = xtan\theta_o - \large\frac{g}{2v_o^2cos^2\theta_o}x^2$