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, $x_0 = 0$:

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

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

(a) Velocity and acceleration:

Velocity: $v(t) = \frac{dx}{dt} = 6t^2 - 18t + 12$

$\boxed{v(t) = 6t^2 - 18t + 12 \text{ m/s}}$

Acceleration: $a(t) = \frac{dv}{dt} = 12t - 18$

$\boxed{a(t) = 12t - 18 \text{ m/s}^2}$

(b) Times when particle is at rest:

Set v(t) = 0: $6t^2 - 18t + 12 = 0$ $t^2 - 3t + 2 = 0$ $(t-1)(t-2) = 0$

$\boxed{t = 1 \text{ s and } t = 2 \text{ s}}$

(c) Position when acceleration is zero:

Set a(t) = 0: $12t - 18 = 0$ $t = 1.5 \text{ s}$

$x(1.5) = 2(1.5)^3 - 9(1.5)^2 + 12(1.5) + 5$ $x(1.5) = 6.75 - 20.25 + 18 + 5$

$\boxed{x = 9.5 \text{ m}}$

(a) Velocity and acceleration at t = 2 s:

$\vec{v}(t) = \frac{d\vec{r}}{dt} = 6t\hat{i} + (4 - 3t^2)\hat{j}$

At t = 2: $\vec{v}(2) = 12\hat{i} + (4 - 12)\hat{j} = 12\hat{i} - 8\hat{j}$

$\boxed{\vec{v}(2) = 12\hat{i} - 8\hat{j} \text{ m/s}}$

$\vec{a}(t) = \frac{d\vec{v}}{dt} = 6\hat{i} - 6t\hat{j}$

$\boxed{\vec{a}(2) = 6\hat{i} - 12\hat{j} \text{ m/s}^2}$

(b) Speed at t = 2 s:

$v = |\vec{v}| = \sqrt{12^2 + (-8)^2} = \sqrt{144 + 64}$

$\boxed{v = \sqrt{208} = 14.4 \text{ m/s}}$

(c) Tangential and normal components:

Unit tangent: $\hat{T} = \frac{\vec{v}}{|\vec{v}|} = \frac{12\hat{i} - 8\hat{j}}{14.4}$

Tangential acceleration: $a_T = \vec{a} \cdot \hat{T} = \frac{(6)(12) + (-12)(-8)}{14.4} = \frac{72 + 96}{14.4}$

$\boxed{a_T = 11.7 \text{ m/s}^2}$

Normal acceleration: $a_N = \sqrt{|\vec{a}|^2 - a_T^2}$

$|\vec{a}| = \sqrt{36 + 144} = 13.4 \text{ m/s}^2$

$a_N = \sqrt{(13.4)^2 - (11.7)^2} = \sqrt{179.6 - 136.9}$

$\boxed{a_N = 6.5 \text{ m/s}^2}$