Position, Velocity, and Acceleration

Fundamental Relationships

Position, velocity, and acceleration are related through calculus:

Velocity as derivative of position: $\vec{v}(t) = \frac{d\vec{r}}{dt}$

Acceleration as derivative of velocity: $\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

Position from velocity (integration): $\vec{r}(t) = \vec{r}_0 + \int_{t_0}^{t} \vec{v}(t') \, dt'$

Velocity from acceleration (integration): $\vec{v}(t) = \vec{v}_0 + \int_{t_0}^{t} \vec{a}(t') \, dt'$

One-Dimensional Motion

For motion along a straight line (x-axis):

Position function: $x(t)$

Velocity: $v_x(t) = \frac{dx}{dt}$

Acceleration: $a_x(t) = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

Example: Polynomial Position Function

If $x(t) = 3t^3 - 2t^2 + 5t + 1$ (in meters), then:

Velocity: $v_x = \frac{dx}{dt} = 9t^2 - 4t + 5 \text{ m/s}$

Acceleration: $a_x = \frac{dv_x}{dt} = 18t - 4 \text{ m/s}^2$

Two-Dimensional Motion

Position vector in 2D: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$

Velocity vector: $\vec{v}(t) = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} = v_x\hat{i} + v_y\hat{j}$

Acceleration vector: $\vec{a}(t) = \frac{dv_x}{dt}\hat{i} + \frac{dv_y}{dt}\hat{j} = a_x\hat{i} + a_y\hat{j}$

Speed (magnitude of velocity): $v = |\vec{v}| = \sqrt{v_x^2 + v_y^2}$

Projectile Motion

For projectile motion with no air resistance:

Horizontal component:

• $a_x = 0$
• $v_x = v_{0x} = v_0\cos\theta$ (constant)
• $x(t) = x_0 + v_{0x}t$

Vertical component:

• $a_y = -g$
• $v_y(t) = v_{0y} - gt = v_0\sin\theta - gt$
• $y(t) = y_0 + v_{0y}t - \frac{1}{2}gt^2$

Variable Acceleration

When acceleration is a function of time, $a(t)$:

$v(t) = v_0 + \int_0^t a(t') \, dt'$

$x(t) = x_0 + \int_0^t v(t') \, dt'$

Example: Time-Dependent Acceleration

If $a(t) = 6t - 4$ m/s², and $v_0 = 2$ m/s, $x_0 = 0$:

$v(t) = 2 + \int_0^t (6t' - 4) \, dt' = 2 + 3t^2 - 4t$

$x(t) = \int_0^t (2 + 3t'^2 - 4t') \, dt' = 2t + t^3 - 2t^2$