Essential Questions:
What variables shape the motion of an object?
How do mathematical descriptions of motion connect to our everyday life?
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.
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Δx = change in position, distance, displacement, length [m, meters]
xi = initial (starting) position [m, meters]
xf = 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.
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Δt = change in time, time period [s, seconds]
ti = initial (starting) time [s, seconds]
tf = final (ending) time [s, seconds]
Velocity
Velocity is a measure of how much a position changes (Δx) over a period of time (Δt).
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Δx = distance [m, meters]
Δt = 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
$$v = \frac{\Delta x}{\Delta t}$$ $$20\frac{m}{s} = \frac{\Delta x}{80s}$$ $$20\frac{m}{s}80s = \Delta x$$ $$1600m = \Delta x$$Example: Someone says 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
$$10 km = 10(1,000)m = 10,000 m$$ $$1h \left(\frac{60min}{1h}\right)\left(\frac{60s}{1min}\right) = 3,600s$$ $$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{10,000m}{3,600s}$$ $$v = 2.7\small\frac{m}{s}$$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
$$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{270miles}{4hour}$$ $$v = 67.5\small\frac{miles}{hour}$$Example: If I walk at a speed of 1.2m/s how long will it take for me to walk 2km?
solution
$$2 km = 2,000 m$$ $$v = \frac{\Delta x}{\Delta t}$$ $$\Delta t = \frac{\Delta x}{v}$$ $$\Delta t = \frac{2,000m}{1.2\small\frac{m}{s}}$$ $$\Delta t = \frac{2,000}{1.2}s$$ $$\Delta t = 1,666.\overline{6} s$$Acceleration
Just like velocity is a change in position over time, acceleration is a change in velocity over time.
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Δv = change in velocity [m/s] = vf - vi
Δt = time period [s, seconds]
a = acceleration [m/s², meters per second squared]
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
$$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{4.43\small\frac{m}{s}}{0.45s}$$ $$a = 9.84\small\frac{m}{s^{2}}$$Example: The internet says Tesla Model S can go from 0 to 60 miles per hour in 5.9s. What is the acceleration in m/s²? (1 mile = 1609 meters)
solution
$$ 60\left(\frac{\color{red}{mile}}{\color{blue}{hour}}\right)\left(\frac{1609 m}{1 \color{red}{mile}}\right)\left(\frac{1 \color{blue}{hour}}{3600s}\right) = 26.8\small\frac{m}{s} $$ $$a = \frac{\Delta v}{\Delta t}$$ $$a = \frac{26.8\small\frac{m}{s}}{5.9s}$$ $$a = 4.5\small\frac{m}{s^{2}}$$Example: I start a velocity of 1 m/s. I speed up to 3 m/s over 10 seconds. What is my acceleration?