Introduction to Kinematics: Describing Motion in Physics

While speed is a scalar (just a number), velocity is a vector, meaning it includes both magnitude and direction.

Acceleration: Changing Velocity

Motion is rarely constant. When an object speeds up, slows down, or changes direction, it is undergoing acceleration ($a$). Acceleration is defined as the rate at which velocity changes over time.

Just as with velocity, we can define average and instantaneous acceleration:

$a_{avg} = \frac{\Delta v}{\Delta t}$ $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$

Note that acceleration is the second derivative of position with respect to time. It is important to remember that an object can have a negative acceleration (deceleration) while still moving in a positive direction, as long as its velocity is decreasing.

Kinematics in One Dimension (1D)

When motion is restricted to a straight line, we use the "Kinematic Equations." These equations are powerful tools, but they come with a strict requirement: they only apply when acceleration is constant.

If $a$ is constant, we can use the following relationships to find position, velocity, or time:

1. $v = v_0 + at$
2. $\Delta x = v_0t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a\Delta x$

Where $v_0$ is the initial velocity and $t$ is the time elapsed.

Worked Example: The Braking Car

A car traveling at $30\text{ m/s}$ slams on its brakes and comes to a complete stop over a distance of $60\text{ meters}$. What was the car's acceleration?

Step 1: Identify knowns and unknowns.

• $v_0 = 30\text{ m/s}$
• $v = 0\text{ m/s}$ (since it stops)
• $\Delta x = 60\text{ m}$
• $a = ?$

Step 2: Choose the correct equation. Since we don't have time ($t$), we use the third equation: $v^2 = v_0^2 + 2a\Delta x$

Step 3: Solve for $a$. $0^2 = (30)^2 + 2(a)(60)$ $0 = 900 + 120a$ $-900 = 120a$ $a = -7.5\text{ m/s}^2$

The negative sign indicates that the car is decelerating.

Multi-Dimensional Motion

In the real world, objects rarely move in perfectly straight lines. When motion occurs in two or three dimensions (such as projectile motion), we treat each dimension independently.

In 2D space, we describe position using vectors: $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$

The velocity and acceleration vectors are also broken down into components: $\vec{v} = v_x\hat{i} + v_y\hat{j}$ $\vec{a} = a_x\hat{i} + a_y\hat{j}$

The key principle here is that the horizontal motion does not affect the vertical motion. For a projectile in free fall (ignoring air resistance), the horizontal acceleration $a_x$ is zero, while the vertical acceleration $a_y$ is equal to the acceleration due to gravity ($g \approx 9.8\text{ m/s}^2$ downward). This independence allows us to solve complex 2D problems by splitting them into two simple 1D problems.