You run around a circular 400m track one time in 15seconds.  What is your velocity?

Zero!, remember the question asks for VELOCITY and not SPEED

You live 15Km from school.  You leave and get half-way there and remember you need to go to the store to pick up your lunch.

You spend 15minutes at the store then head right to school.  Assume your speed is 70Km/Hr the whole time.
What is your velocity on this trip?

Remember that you need displacement and time to get velocity.
The final displacement is easy: 15Km
Now we need the time.  To the time, we must consider all the parts of the trip...
 
from house to turn-around t=(7.5Km)/(70K/H) .107 Hr
from turn-around to store t=(6+7.5Km)/(70Km/H) .193 Hr
time at store 15min*(1Hr/60min) .25 Hr
store to school t=(21Km)/(70Km/H) .3 Hr

So our total time is .85 Hr.
We will call the home our zero point and the school is in the + direction.
The formula for velocity is (Xf-Xo)/t, 15Km-0Km/.85Hr=17.65Km/Hr

NOTE: since all motion was at 70k/h, we could have just added 7.5+7.5+6+6+15=42Km,
then 42k/70k/hr=.6Hr.  Then add in the .25Hr of store time: .25+.6=.85Hr

You know that your velocity on a trip is -2m/s and that the trip took 24seconds.  If you started at a point given by +500m, then what is your final position?

We know that V=Xf-Xo/t, so Xf must equal Vt+Xo... (-2m/s)24s+(+500m)=+452m

You get on a bus and start walking to the back.  You are eating a sandwich.  A bug is on the sandwich.  The bug wants to escape, so is also heading toward the back of the bus.
Given these velocities, what is the net velocity of the bug?
BUS= +2m/s
YOU= -1m/s
SANDWICH= +.05m/s
BUG= -.25m/s
The bug's velocity depends on all the others, so we just add them all together....
(+2)+(-1)+(+.05)+(-.25)= +.8m/s, with the "+" meaning the same direction as bus motion
note: this answer would be the view of an observer standing outside the bus, watching you.

According to the bus driver, what is the velocity of the sandwich?
Since the bus driver thinks he is stationary, he sees the sandwich's velocity depend on 2 things:
the motion of the person eating it, and the motion of the sandwich giong into your mouth.
(-1)+(+.05)= -.95m/s, with the "-" meaning toward the back of the bus
Think about it.  If you're in a car with your friends zooming down the road at 60mi/hr, as far as you're concerned, your velocities are all zero inside the car... you can reach over and touch them.  If someone passes you at 65mi/hr, they would look to you like they were going just a little faster, YET to someone on the sidewalk, both cars are going pretty fast!

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