MOTION REVIEW 2
Instantaneous Velocity
So we’ve reviewed the basics of vectors and scalars, and the relationship between
position, velocity, and acceleration. But so far we’ve only covered average velocities, where the displacement over time would give us the answer. When we’re
working with more complicated motion, like obejcts that are falling or projectiles
being thrown or propelled through the air, we need to be more exact about velocity
at a certain time. For example: average velocity won’t help us if we’re asked how
fast a projectile is falling when it hits the ground. For that we need to use our
Kinematics equations. Here they are, for your review:
vf 2 = vi 2 +2ad
d = vi + ½at 2
(final velocity squared equals
(displacement equals initial velocity
initial velocity squared plus two times plus one half of acceleration times time
acceleration times displacement)
squared)
vf = vi + at
(final velocity equals initial velocity
plus acceleration times time)
d=
( v 2+ v ) t
f
i
(displacement equals the sum of final
velocitiy and initial velocity divided by
two times time)
All of these equations can be useful when solving problems about objects that
have a constant acceleration in one direction. This means anything that is in a
state of free fall (gravity is a constant acceleration), whether it is dropped, thrown,
launched, etc. It can also be any other problem where acceleration is assumed to
be constant, and motion is linear. Note that between all four equations, there are
only four variables used: Displacement, Velocity, Acceleration, and Time. You
might also note that no equation uses more than three of those variables in total.
So depending on the information a problem might give us, we can choose which
equations to use. For example: if we know an object is dropped and falls for ten
seconds, which equation might we choose to solve for its final velocity?
How about vf = vi + at?
Given that we know its initial velocity
(zero), and its acceleration (gravitational
acceleration = -9.8m/s2), can we solve for
its final velocity?
Once we’ve done that, could we now
solve for the distance it travels while falling (put another way: its displacement)?
Which equation would we use for that?
Parabolic Motion
Sometimes, projectiles are not just moving in one direction (like a falling object
moves just up and down). In these cases, multiple vectors moving different directions must be added together to describe the motion of the object.
For instance: Let’s imagine a ball being thrown by a pitcher. This pitcher has had
amazing analytics applied to her throw and knows how fast the ball leaves her
hand traveling both horizontally (we’ll call this dimension “x”) and vertically (we’ll
call this dimension “y”). So when she throws the ball, she knows it is traveling 30
m/s horizontally and 5 m/s vertically.
vector breakdown
5 m/s “y”
vector
path of the ball’s flight
r
vecto city
o
tant
resul ball’s vel
e
h
t
f
o
y
Pitcher’s
mound
30 m/s - “x” vector
x
As you can see, the flight of the ball follows a parabolic path. This results from the
constant acceleration due to gravity in the “y” dimension and the constant velocity of the projectile in the “x” dimension. Put another way: the ball falls up and
down due to gravity at a constant rate of acceleration, while it travels from side to
side at a constant velocity.
This means we can use our linear motion equations separately on each component vector of the object’s trajectory. For example: how long would the ball be in
the air if it is caught at the same height it is thrown? (Hint: use your kinematics
equations on the “y” dimension of the ball’s flight.) How high in the air is the
peak of its flight?
Once you know that, how far does the ball travel horizontally over the course of
its flight? (Hint: use the simple velocity = distance over time equation on the “x”
dimension of the ball’s flight)
path of the ball’s flight
solve for the question marks
y
?
Pitcher’s
mound
?
x
MOTION PRACTICE QUESTIONS
1. A projectile is launched straight up in the air at a velocity of 100 meters per second.
a. How high in the air will it travel?
b. How long will the projectile spend in the air?
c. What will the speed of the projectile be when it hits the ground?
2. A stunt driver drives her camero off the side of a cliff moving east at a speed of
40 m/s and starts a stopwatch. When she hits the ground, the stopwatch reads 20
seconds.
a. How far east of the cliff is the impact point?
b. What is the height of the cliff?
c. What is the vertical speed of the car when it hits the ground?
3. A bullet is fired at 100 m/s straight down into a ballistic foam specifically designed to slow the bullet down at a constant rate (assume a constant acceleration).
The bullet travels 5 meters into the foam before it stops.
a. What is the acceleration of the bullet in the foam?
b. How long does it take the bullet to reach a complete stop?