It is important to know where things are in both space and time. Physics looks not only at position, but how position changes as time changes.

# Position

In physics we use the symbol x for position. That means you shouldn't use x for an unknown unless you are solving for position. We generally use meters(m) as our units.

To represent a distance we subtract the final position from the initial position. We use the Δ symbol to show change.

# $$ \Delta x = x_{f}-x_{i}$$

\(\Delta x\) = change in position, distance, displacement, length [m, meters]

\(x_i\) = initial (starting) position [m, meters]

\(x_f\) = final (ending) position [m, meters]

# Time

We keep track of time in the same way as position. We use t for time and generally use seconds(s) as our units.

# $$\Delta t = t_{f}-t_{i}$$

\(\Delta t\) = change in time, time period [s, seconds]

\(t_i\) = initial (starting) time [s, seconds]

\(t_f\) = final (ending) time [s, seconds]

# Velocity

Velocity is a measure of how much position changes (Δx) over a period of time (Δt).

# $$v = \frac{\Delta x}{\Delta t}$$

\(\Delta x\) = distance [m, meters]

\(\Delta t\) = time period [s, seconds]

\(v\) = average velocity [m/s, meters per second]

**Example:**A car is traveling at 20 m/s for 80 seconds. How far does the car travel?

## solution

$$ v = \frac{\Delta x}{\Delta t} $$ $$ v \Delta t= \Delta x$$ $$ (20 \mathrm{\tfrac{m}{s}}) (80\, \mathrm{s}) = \Delta x $$ $$ 1600\, \mathrm{m} = \Delta x $$**Example:**Someone tells you they can run a 10 km race in about an hour. What velocity is that in m/s? Is that fast? (1 m/s is walking speed)

## solution

$$10 \, \mathrm{km} = 10(1,000) \mathrm{m} = 10,000 \, \mathrm{m}$$ $$1\, \mathrm{h} \left(\frac{60\, \mathrm{min}}{1\, \mathrm{h}}\right)\left(\frac{60\, \mathrm{s}}{1\, \mathrm{min}}\right) = 3,600 \,\mathrm{s}$$ $$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{10,000\, \mathrm{m}}{3,600\, \mathrm{s}}$$ $$v = 2.7\, \mathrm{\tfrac{m}{s}}$$**Example:**Google maps says Las Vegas is 4 hours away from Los Angeles. Google says it is 270 miles away. How fast does google think I will drive? Answer this one in miles/hour.

## solution

$$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{270\, \mathrm{miles} }{4\, \mathrm{hour}}$$ $$v = 67.5\, \mathrm{\tfrac{miles}{hour} }$$The speed limit for most of that trip is 70 mph. Google is assuming that everyone is under the speed limit.

**Example:**If I walk at a speed of 1.2 m/s how long will it take for me to walk 2 km?

## solution

$$2 \, \mathrm{km} = 2(1000) \mathrm{m} = 2,000 \, \mathrm{m}$$ $$v = \frac{\Delta x}{\Delta t}$$ $$\Delta t = \frac{\Delta x}{v}$$ $$\Delta t = \frac{2,000\, \mathrm{m}}{1.2\, \mathrm{\tfrac{m}{s}}}$$ $$\Delta t = \frac{2,000}{1.2}\, \mathrm{s}$$ $$\Delta t = 1,666.\overline{6} \, \mathrm{s}$$# Acceleration

Acceleration is a measure of how much velocity changes (Δv) over a period of time (Δt).

# $$a = \frac{\Delta v}{\Delta t}$$

\(\Delta v\) = change in velocity [m/s] = \(v_f-v_i\)

\(\Delta t\) = time period, change in time [s, seconds]

\(a\) = acceleration [m/s²]

**Example:**A basketball falls off a table and hits the floor in 0.45 s. The ball has a velocity of 4.43 m/s right before it hits the ground. What is the acceleration of the basketball as it falls?

## solution

$$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{4.43\, \mathrm{\tfrac{m}{s}}} {0.45\, \mathrm{s}}$$ $$a = 9.84\, \mathrm{\tfrac{m}{s^{2}} }$$**Example:**A car can go from 0 to 60 miles per hour in 5.9 s. What is the acceleration in m/s²? (1 mile = 1609 meters)

## solution

$$ 60\left( \mathrm{ \frac{\color{red}{mile}}{\color{blue}{hour}}} \right)\left(\frac{1609\,\mathrm{ m}}{1 \,\color{red}{\mathrm{mile} }}\right)\left(\frac{1\, \color{blue}{ \mathrm{hour} }}{3600\, \mathrm{s} }\right) = 26.8 \mathrm{\tfrac{m}{s}} $$ $$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{26.8\,\mathrm{ \tfrac{m}{s}}} {5.9\, \mathrm{s}}$$ $$a = 4.5\, \mathrm{ \tfrac{m}{s^{2}} }$$**Example:**I start a velocity of 1 m/s. I speed up to 3 m/s over 10 seconds. What is my acceleration?

## solution

$$\Delta v = v_{f}-v_{i}$$ $$\Delta v = 3 \,\mathrm{ \tfrac{m}{s}} -1\, \mathrm{ \tfrac{m}{s} }$$ $$\Delta v = 2\, \mathrm{\tfrac{m}{s}}$$$$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{2\, \mathrm{\tfrac{m}{s}} }{10\, \mathrm{s} }$$ $$a = 0.2\,\mathrm{ \tfrac{m}{s^{2}}}$$

**Example:**These graphs show velocity vs. time. What do their slopes represent?

## solution

acceleration

acceleration is defined as the change in velocity every second. This is the same as the slope of the graphs.