General Physics I: Day 5
Circular Motion and Relative Velocity (kinda)
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The diagram below shows two successive positions
of a particle; it’s a segment of a full motion
diagram. Which of the acceleration vectors best
represents the acceleration between ti and tf?
Projectile Motion
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Projectiles: Objects acted on only by gravity, free to
move horizontally as well as vertically.
Keys to analyzing projectile motion:
• Each component of motion is independent!
You must separate horizontal and vertical.
• Treat each component with the equations for
constant acceleration.
• Always ignore air resistance unless specifically
directed otherwise.
Projectile Motion: y-component
Just like 1D free-fall motion…
a y   g   9.8 m s
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• Time in the air is determined by the vertical
• 𝑣𝑦 = 0 at the apex of the motion.
• Helpful symmetry! Same height → same 𝑣𝑦
• Horizontal and vertical components share one
piece of information: time!
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Projectile Motion: x-component
In the x-direction there is no acceleration.
ax  
So, of course, horizontal velocity does not change!
v f ,x  vi ,x
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Warm-Up: Cannonballs!
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Two cannonballs, A and B, are fired from level
ground with identical speeds, but with A fired at a
larger angle relative to the ground than B.
What can you say about the hang A > B → ~83%
times of the two cannonballs?
B > A → ~8%
A = B → ~8%
What can you say about how
far the two cannonball travel?
A > B → ~25%
B > A → ~33%
A = B → ~8%
Depends → ~33%
Warm-Up: Cannonballs!
“A.) A will spend more time in the air due to the
increased angle it was fired at, while B will hit the
ground quicker because it was fired closer to the
ground. B.) A will go further while B will travel a
shorter distance.”
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Warm-Up: Cannonballs!
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“A. with cannonball A shot at a larger angle than B,
its height will be much higher than B so it will have
a longer hang time.
B. as stated in part a, cannonball A was fired at a
larger angle so the cannonball will not travel as far
as cannonball B. The height of A will be much
higher than B, but because the angle of cannonball
B is smaller, it will be able to travel father than A.”
1st part right, but not the 2nd (very common)
Warm-Up: Cannonballs!
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“The angle of the launch affects the distance. A ball
shot at 88 degrees and one at 2 degrees will both not go
very far compared with one shot at 60 degrees.”
“b. A 45 degree angle is the optimal angle when firing
the cannonballs to reach the maximum distance (when
all other variables are constant). The question does not
specify the angles at which either A or B were fired at
so I can't determine which one would travel farther but
it would be whichever one was fired at an angle closest
to 45 degrees above the ground.”
Warm-Up: Cannonballs!
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Use extremes (a.k.a “limiting cases”) when tackling
things like this:
• What if A is fired at 89° and B is fired at 50°?
• What if A is fired at 30° and B is fired at 1°?
(angles relative to the horizontal)
Warm-Up: Cannonballs!
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Two cannonballs, A and B, are fired from level
ground with identical speeds, but with A fired at a
larger angle relative to the ground than B.
What can you say about the hang A > B → ~83%
times of the two cannonballs?
B > A → ~8%
A = B → ~8%
What can you say about how
far the two cannonball travel?
A > B → ~25%
B > A → ~33%
A = B → ~8%
Depends → ~33%
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A battleship simultaneously fires two shells at target
dummies. All we know is that the shells follow the
ideal parabolic trajectories shown, which target gets
hit first?
A) A
B) B
C) Both at the same time
D) Need more information
Worked-Example: Flour Bomb
Where to release?
How fast (x & y) is it
going when it hits?
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Worked-Example: Flour Bomb
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1. What would have changed if we had placed the
origin on the ground directly below where the flour
was released? The answers? The equations? Any of
the values we found? Explain.
~14% → The answers would change
~70% → Some #s in the calculation,
but not the answers
~14% → Stuck or didn’t answer this question
Worked-Example: Flour Bomb
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Worked-Example: Flour Bomb
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2. If the flour bag had been shot downward at 10
m/s (instead of just being released), how would that
have changed the final vertical speed of the bag?
How would it have changed the final velocity?
~70% → 𝑣f,𝑦 would be larger
~42% → 𝑣f would be larger
~14% → 𝑣f would be the same
~14% → 𝑣f would be smaller
Worked-Example: Flour Bomb
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A small ball rolls horizontally off the edge of a
tabletop that is 1.2 m high. It strikes the floor at a
point 1.52 m horizontally from the edge of the
table. How long is the ball in the air?
A) t = 0.20 s
B) t = 0.50 s
C) t = 0.56 s
D) t = 0.63 s
Follow up: What was its speed as it left the table?
Warm-Up: Slingshot Speed
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A girl wishes to figure out the maximum speed at
which her slingshot fires rocks, but she has no access
to timing devices of any kind. Using your knowledge
of projectile physics, describe how she could do this
using nothing but a good measuring tape.
~17% →
~8% →
~25% →
~50% →
Good “lone wolf” method
Good “it takes a village” method
Really wanted to measure time or angle
Wrong track or stuck
Warm-Up: Slingshot Speed
“She first has to measure the angle she is shooting
the slingshot at so she can take the cosine and sine
and multiply it to the distance the rock travels to
find Vx and Vy respectively.
[Equations V0x=V0cos(theta)0 and Vx=V0x]
She can then take the square root of Vx^2 + Vy2
which will give her the speed.”
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Warm-Up: Slingshot Speed
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“[…] I would tell her to count the number of
seconds the best she could […]”
“[…] make sure to count the time it took for the
rock to reach it's landing area […]”
Many of you have great confidence in a kid’s ability
to time things accurately by counting out seconds!
Warm-Up: Slingshot Speed
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“Since gravity is a known value, have the girl
launch the rock vertically at 90 degrees and
measure max height, then calculate velocity need to
reach that height.”
This works, but measuring the maximum height is
not exactly easy!
Warm-Up: Slingshot Speed
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“By laying out the measuring tape, it is possible to
test the maximum speed of the rocks by pulling the
slingshot back as far as possible, and then firing at
different angles, until an angle is found at which the
rocks fly the furthest.”
This will work, but takes a lot of repetition.
Warm-Up: Slingshot Speed
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“Fire the rock parallel to the ground. Measure the
height of the rock from the ground before it it fired
(y). Measure the distance it travels till hit first hits
the ground (x). Plug y into this equation -y=(1/2)(9.8)t^2 to solve for time (t). Plug t and x into this
equation x=Vo*t to solve for the initial velocity
when it leaves the sling.”
Yes!
If you shoot horizontally, the vertical drop acts as
the timer!
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One ball is going to be dropped from the platform
while simultaneously another is launched
horizontally from the platform. They both then fall
to the ground. Which ball hits first?
A) The dropped ball.
B) The horizontally launched ball.
C) They hit at the same time.
D) Depends on the speed of the launched ball.
E) Depends on their masses.
Acceleration Without Changing Speed
A ball swings in a circle at constant speed.
Is it accelerating? Yes!
Can its acceleration be directed in the same
direction as its velocity? No!
Can its acceleration be directed in the opposite
direction as its velocity? No!
So the 𝑎 must be perpendicular to the velocity…
Inward! An object in circular motion has inward
acceleration called centripetal acceleration.
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Centripetal Acceleration
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Centripetal acceleration always points towards the
center of the circle
ac 
v
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What matters is speed
r
and turning radius:
If you only go faster, acceleration is larger.
If you only tighten your turn, acceleration is larger.
Centripetal acceleration is the amount of acceleration
required for an object to go in a circle of radius r at
speed v.
WarmUp: Ranking Circles
Rank from largest to smallest radial acceleration.
~14% → B = E > A = C = D
~42% → B > A = C = E > D
~42% → B > E > A = C > D
~2% → C = E > A = B = D
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Which of the following best describes the
magnitude of the acceleration that you are
experiencing around the axis of the earth’s rotation
right now?
A) Your acceleration is greater than g.
B) Your acceleration is equal to g.
C) Your acceleration is less than g.
D) Your acceleration is zero.
Coming up…
Thursday (9/4) → Catch up & Ch 4 Mini-Exam
Tuesday (9/9) → Ch 4 & Mini-Exam
Mini-Exam bumped to next Tuesday
No Warm-Up for Thursday. Study & practice!
Mastering Hwk #3 due Sunday by 11:59 PM
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