Motion: Review

Metric Units

Physical quantities can be expressed in terms of these metric base units:

distance: meter (m)
time: second (s)
mass: kilogram (kg)

We use metric prefixes to indicate multiplication or division by powers of ten. Converting from 10km to meters means moving the decimal 3 spaces larger.

$$10km = 10\left(10^{3}\right)m = 10{,}000m$$ Converting from 10ms to meters means moving the decimal 3 spaces smaller. $$10 ms = 10\left(10^{-3}\right)s = 0.01s$$
Name Symbol Factor Power
giga G 1000 000000 109
mega M 1000 000 106
kilo k 1000 103
centi c 0.01 10−2
milli m 0.001 10−3
micro μ 0.000001 10−6
nano n 0.000000 001 10−9
Example: 10km is how many meters?
solution

replace k with 1000

$$10km = 10(1000)m = 10{,}000m$$
Example: 450nm is how many meters?
solution

replace n with 10-9

$$450nm = 450\left( 10^{-9}\right)m = 0.00000045m$$

Nonmetric Units

Converting outside the metric system is more complex. You can do a google search to find out how two units are related. You then build a conversion fraction with the new unit on top and the old unit on the bottom to be canceled out. The top and the bottom of the fraction must be equal to each other.

Lets convert 50 minutes into seconds.

$$ 50 min \to s$$ In order to cancel minutes we want to build a fraction with minutes on top and the seconds on the bottom. $$ 50 min \left( \frac{s}{min} \right)$$ The fraction must be equal to one to not change the value we are converting. This means the top and bottom must equal each other. One minute equals 60 seconds. $$ 50 min \left( \frac{60s}{1min} \right)$$ We can cancel units. $$ 50 \color{red}{min} \left( \frac{60s}{1 \color{red}{min} } \right)$$ $$ 50 \left( \frac{60s}{1} \right)$$ Multiply and divide to simplify $$3000s$$
Example: A marathon is 26.2 miles. If one mile is the same as 1.6 km, how far is a marathon in kilometers?
solution $$26.2 mile \left( \frac{1.6 km}{1 mile} \right) = 41.92km$$
Example: I'm 6 feet and 1 inch tall. How many meters tall am I?
solution

Let's get everything in inches first

$$6ft\left( \frac{12inch}{1ft}\right) + 1inch$$ $$72inches + 1inch$$ $$73inches$$

Now we convert into meters.
A google search says that 1m = 39.37inches

$$73inches \left( \frac{1m}{39.37inch}\right) $$ $$73 \left( \frac{1m}{39.37}\right) $$ $$1.85m$$

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 $$\begin{aligned} v &= \small\frac{\Delta x}{\Delta t} \\ \\ 20\tfrac{m}{s} &= \small\frac{\Delta x}{80s} \\ \\ 20\tfrac{m}{s}80s &= \Delta x \\ \\ 1600m &= \Delta x \end{aligned}$$
Example: Someone tells you they can run a 10km race in about an hour. What velocity is that in m/s? Is that fast? (1m/s is walking speed)
solution $$10 km = 10(1,000)m = 10,000 m$$ $$1h \left(\frac{60min}{1h}\right)\left(\frac{60s}{1min}\right) = 3,600s$$ $$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{10,000m}{3,600s}$$ $$v = 2.7\tfrac{m}{s}$$
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 on average? Answer this one in miles/hour.
solution $$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{270miles}{4hour}$$ $$v = 67.5\tfrac{miles}{hour}$$
Example: If I walk at a speed of 1.2m/s how long will it take for me to walk 2km?
solution $$2 km = 2(1000)m = 2,000 m$$ $$v = \frac{\Delta x}{\Delta t}$$ $$\Delta t = \frac{\Delta x}{v}$$ $$\Delta t = \frac{2,000m}{1.2\small\frac{m}{s}}$$ $$\Delta t = \frac{2,000}{1.2}s$$ $$\Delta t = 1,666.\overline{6} s$$

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 0.45s. The ball has a velocity of 4.43m/s right before it hits the ground. What is the acceleration of the basketball as it falls?
solution $$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{4.43\small\frac{m}{s}}{0.45s}$$ $$a = 9.84\tfrac{m}{s^{2}}$$
Example: A car can go from 0 to 60 miles per hour in 5.9s. What is the acceleration in m/s²? (1 mile = 1609 meters)
solution $$ 60\left(\frac{\color{red}{mile}}{\color{blue}{hour}}\right)\left(\frac{1609 m}{1 \color{red}{mile}}\right)\left(\frac{1 \color{blue}{hour}}{3600s}\right) = 26.8\small\frac{m}{s} $$ $$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{26.8\small\frac{m}{s}}{5.9s}$$ $$a = 4.5\tfrac{m}{s^{2}}$$
Example: I start a velocity of 1 m/s. I speed up to 3 m/s over 10 seconds. What is my acceleration?
solution $$\Delta v = v_{f}-v_{i}$$ $$\Delta v = 3\small\frac{m}{s}-1\small\frac{m}{s}$$ $$\Delta v = 2\small\frac{m}{s}$$
$$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{2\small\frac{m}{s}}{10s}$$ $$a = 0.2\tfrac{m}{s^{2}}$$
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.

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