4.2 Acceleration Vector

Acceleration vectors are key to understanding how objects move and change speed. They show us how velocity changes over time, helping us predict an object's path and future position.

By breaking down acceleration into components, we can tackle complex motion problems. This approach lets us analyze movement in multiple dimensions, making it easier to solve real-world physics challenges.

Acceleration Vector

Acceleration vector calculation

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• Acceleration represents rate of change of velocity over time
• Velocity in unit vector notation expressed as $\vec{v}(t) = v_x(t)\hat{i} + v_y(t)\hat{j} + v_z(t)\hat{k}$
  • $v_x(t)$, $v_y(t)$, $v_z(t)$ represent component functions of velocity in $x$, $y$, $z$ directions
  • $\hat{i}$, $\hat{j}$, $\hat{k}$ represent unit vectors in $x$, $y$, $z$ directions
• Calculate acceleration vector by differentiating each velocity vector component with respect to time (time derivative)
  • $\vec{a}(t) = \frac{d\vec{v}(t)}{dt} = \frac{dv_x(t)}{dt}\hat{i} + \frac{dv_y(t)}{dt}\hat{j} + \frac{dv_z(t)}{dt}\hat{k}$
  • Resulting vector $\vec{a}(t)$ represents acceleration vector in unit vector notation
• Example: Given $\vec{v}(t) = (3t^2)\hat{i} + (2t)\hat{j} + (4)\hat{k}$, find $\vec{a}(t)$
  • $\vec{a}(t) = \frac{d}{dt}((3t^2)\hat{i}) + \frac{d}{dt}((2t)\hat{j}) + \frac{d}{dt}((4)\hat{k}) = (6t)\hat{i} + (2)\hat{j} + (0)\hat{k}$

Particle motion under constant acceleration

• Constant acceleration causes particle's velocity to change at constant rate
• Acceleration vector remains constant in both magnitude and direction
• Determine particle's position using kinematic equations for constant acceleration
  • $\vec{r}(t) = \vec{r}_0 + \vec{v}_0t + \frac{1}{2}\vec{a}t^2$
    • $\vec{r}(t)$ represents position vector at time $t$
    • $\vec{r}_0$ represents initial position vector
    • $\vec{v}_0$ represents initial velocity vector
    • $\vec{a}$ represents constant acceleration vector
  • $\vec{v}(t) = \vec{v}_0 + \vec{a}t$
    • $\vec{v}(t)$ represents velocity vector at time $t$
• Apply equations to each component of position and velocity vectors independently
• Example: Particle starts at $\vec{r}_0 = (0)\hat{i} + (0)\hat{j} + (0)\hat{k}$ with $\vec{v}_0 = (2)\hat{i} + (3)\hat{j} + (1)\hat{k}$ and $\vec{a} = (1)\hat{i} + (2)\hat{j} + (0)\hat{k}$. Find $\vec{r}(3)$ and $\vec{v}(3)$
  • $\vec{r}(3) = (0 + 2(3) + \frac{1}{2}(1)(3^2))\hat{i} + (0 + 3(3) + \frac{1}{2}(2)(3^2))\hat{j} + (0 + 1(3) + \frac{1}{2}(0)(3^2))\hat{k} = (10.5)\hat{i} + (18)\hat{j} + (3)\hat{k}$
  • $\vec{v}(3) = (2 + 1(3))\hat{i} + (3 + 2(3))\hat{j} + (1 + 0(3))\hat{k} = (5)\hat{i} + (9)\hat{j} + (1)\hat{k}$

Constant acceleration in multiple dimensions

• Apply one-dimensional motion equations for constant acceleration to each component of motion in three dimensions
  • $x(t) = x_0 + v_{0x}t + \frac{1}{2}a_xt^2$
  • $y(t) = y_0 + v_{0y}t + \frac{1}{2}a_yt^2$
  • $z(t) = z_0 + v_{0z}t + \frac{1}{2}a_zt^2$
• Treat each component of motion independently as one-dimensional problem
• Find final position, velocity, or acceleration by combining components using vector addition
  • $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$
  • $\vec{v}(t) = v_x(t)\hat{i} + v_y(t)\hat{j} + v_z(t)\hat{k}$
  • $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$
• Example: Particle starts at $(2, 3, 1)$ with initial velocity $(1, -2, 0)$ and acceleration $(0, 1, -1)$. Find position after 4 seconds
  1. $x(4) = 2 + 1(4) + \frac{1}{2}(0)(4^2) = 6$
  2. $y(4) = 3 + (-2)(4) + \frac{1}{2}(1)(4^2) = 7$
  3. $z(4) = 1 + 0(4) + \frac{1}{2}(-1)(4^2) = -7$
  • $\vec{r}(4) = (6)\hat{i} + (7)\hat{j} + (-7)\hat{k}$

Unit vector notation for acceleration

• In two dimensions, acceleration represented as $\vec{a} = a_x\hat{i} + a_y\hat{j}$
  • $a_x$, $a_y$ represent components of acceleration in $x$, $y$ directions
• In three dimensions, acceleration represented as $\vec{a} = a_x\hat{i} + a_y\hat{j} + a_z\hat{k}$
  • $a_x$, $a_y$, $a_z$ represent components of acceleration in $x$, $y$, $z$ directions (vector components)
• Calculate magnitude of acceleration vector using Pythagorean theorem
  • Two dimensions: $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$
  • Three dimensions: $|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$
• Describe direction of acceleration vector using angles with respect to coordinate axes or unit vector components
• Example: Acceleration vector $\vec{a} = (3)\hat{i} + (-4)\hat{j}$
  • Magnitude: $|\vec{a}| = \sqrt{3^2 + (-4)^2} = 5$
  • Direction: $\tan\theta = \frac{-4}{3}$, $\theta \approx -53.1°$ from positive $x$-axis

Acceleration in different reference frames

• Acceleration can be described relative to different reference frames
• Relative motion between reference frames affects observed acceleration
• Vector calculus techniques are used to transform acceleration between reference frames