Acceleration is another one of those palabras that spends time in the realm of the everyday lexicon.
Go faster!! Speed up!! Slow down!! Turn left!!
Each of the above phrases involves an acceleration which essentially is a change in velocity. So we are moving from constant velocity to a change in speed or a change in direction.
This is very important!!
Acceleration is a vector and, just like velocity, it has a magnitude and a direction. So a change in direction is an acceleration the same way an increase or decrease in speed is an acceleration. Therefore, in our vehicles, there are three accelerators, (a) the gas pedal (b) the brake (c) the steering wheel.
So the position 🆚 time graph for an accelerating object is not a straight line. Why?!?! It’s because the slope is not constant as velocity is changing! So let’s delve!! Here is a motion analysis for Jaylene on her scooter going down Pine St.
🚨So we have come up to a rule that will help you throughout the course.🚨
When an object, moving in a straight line, speeds up, its velocity and acceleration have the same sign (+,+ or -,-). When an object moving in a straight line slows down, its velocity and acceleration have opposite signs (+,- or -,+).
So let’s take it up a notch and analyze a world record frisbee catch by one amazing puppy!!
In taking some liberties, here is the golden analysis I came up with and it’s really incredible!!
THE KINEMATIC EQUATIONS
The link above ⬆️ will take you to everything you need to know about a special set of equations that apply to motion with constant acceleration, including some very important examples. Like we said, most acceleration in nature is constant.
FREE FALL!!
🚨 Super Important NOTES Part Deux 🚨
And perhaps the most important case of constant acceleration is the motion of objects near the surface of the earth. That is, free fall occurs when only the pull of gravity affects an object’s fall with a constant downward acceleration. Again, it is vital to understand all of the examples and physics speak! Ask questions!
This acceleration due to gravity, g, is the same whether the object is moving up, moving down or even momentarily at rest when it switches directions at its maximum height. Do you think mass affects this acceleration??
Many real life situations come close to this ideal like a high diver descending into the pool or a bullfrog in midair. With relatively low speed, there is minimal air resistance and definitive free fall.
If a falling object, at high speed, has a large cross section, we cannot ignore air resistance. In this case, the motion is no longer free fall because acceleration is not constant. In nature, a hawk plummeting at high speeds reaches its terminal velocity at which the effects of air resistance balances the downward pull of gravity, and therefore can no longer accelerate.
With objects in free fall, we can use the Kinematic Equations in the vertical direction with 9.8 m/s^2 as the constant acceleration due to gravity, or g. On many occasions, it is perfectly acceptable to round to 10 m/s^2 especially in multiple choice situations. So, in the vertical direction, if something is dropped, initial velocity and initial position are both zero, then vertical displacement equation becomes y = 1/2(g)t^2.
The Acceleration due to Gravity or “little g”
The value of g at sea level on earth differs from the acceleration due to gravity on other planets, spacecrafts and even on Mt. Everest!
This is the acceleration due to gravity that affects objects moving upwards, downwards and objects momentarily at rest!!
If we consider upward to be positive and Terry tosses a tennis ball straight up, it has a positive velocity and a negative acceleration. According to our rule, if acceleration and velocity have opposite signs, the ball slows down until the highest point when v = 0 m/s as it stops momentarily to change direction. Acceleration cannot be zero at this point or else the ball would just stop in midair!!! As the ball descends, velocity and acceleration are both negative and the ball speeds up as it falls!!
Projectile Motion
Projectile Motion is the amalgamation of constant horizontal velocity and free fall! In essence, the combination of the ideas that comprise the two sets of notes presented above!! The only force acting on a projectile is the downward pull of gravity!! Here is the Golf Lab and DERIVED EQUATIONS for maximum height of the ball flight and total horizontal displacement and these are universal. This is a great introduction to how we manipulate equations to develop relationships between variables that we are given!!
The water fountain and the leaping ballet dancer each follow a curved path so that their direction is constantly changing. Therefore, we can say that they are accelerating at all points in their motion! But this acceleration is only in the vertical direction so horizontal motion is constant.
For projectile motion, we will always place an axis along the direction of the acceleration due to gravity. Therefore, we can make the direction of the horizontal component of velocity the other axis. So projectile motion is two dimensional motion; motion in a plane. When the ballet dancer leaps, her velocity vector has a horizontal component to the right that remains constant. She also has a velocity vector that points upward which changes due to the acceleration of gravity pulling downwards. An important feature is that the dancer will slow down to zero at her maximum height as quickly as she will speed up on the way down from maximum height. Therefore, corresponding heights in her parabolic arc will have will have the exact same velocity, only a different direction!!!
Let’s play Get the Concept for the changing velocity of a projectile!!
Q: An object launched at some initial speed and angle follows a parabolic arc. At what point during the trajectory is the magnitude of its velocity the smallest? At what point, if any, will the velocity of the object be zero?
A: The magnitude of the velocity is smallest at the peak of its motion. The horizontal component of velocity is constant throughout the trajectory. At maximum height, the vertical (y) component of the velocity is zero (0), so regardless of the magnitude of the horizontal (x) component of velocity, the magnitude of the velocity vector must be the smallest. If the (x) component of velocity is zero, then the tota velocity, both magnitude and direction, is zero (0) at the peak. Because the (x) component of the velocity does not change, horizontal velocity can only be zero if the object was launched straight up!!
So, what do we know about projectile motion?
(1) At maximum height, vertical velocity v(y) is equal to zero.
(2) At maximum height, time (t) is equal to half the total flight time.
(3) Horizontal velocity, v(x), is constant because there is zero (0) horizontal acceleration.
(4) Vertical velocity decreases on the way up, increases on the way down at the same constant rate of 9.8 m/s^2. Sometimes we will round this value to 10 m/s^2 and refer to it as “g”. “g” is always a positive number but generally we call downward the “negative direction”. Therefore, acceleration in the vertical direction a(y) = -g.
(5) We resolve the initial velocity into its vector components and treat both the horizontal and vertical directions ((x) and (y)) as superstar individual problems that we know how to solve using the kinematics equations!!
(6) Time (t) is the same for both directions and cannot be negative and is the bridge between directions.
(7) Since “g” slows things down at the same rate as it speeds things up, vertical velocity (v(y)) is the same at mirroring vertical positions and times during projectile flight.
(8) The magnitude of total velocity is smallest at the peak of motion because the (y) component is zero. If the (x) component is zero, then total velocity is zero. This is a scenario that occurs when a ball is thrown straight up!
(9) We derived the universal derived equations that can be applied to projectile motion!!
FACTOR OF CHANGE!!
Two golf balls are hit from the same point on a flat field. Both are hit it an angle of 30 degrees above the horizontal. Ball 2 has twice the initial velocity of ball 1. Balls 1 and 2 land a displacement d(1) and d(2), respectively, from the initial point. Predict the relationship between the displacements d(1) and d(2), neglecting air resistance.
We will venturing trough these scenarios from the AP workbook in the near future!
Film Clas 