Motion

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.