Motion with Constant Velocityxf = x0 + vt

Motion with Constant Accelerationvf = v0 + atxf = x0 + v0t + at2vf2 = v02 + 2axEquations of Motion!#2 can also be written as x = v0t + at2, since x = xf x0 tThe big three

AP Level Kinematics ProblemA rock is dropped off a cliff and falls the first half of the distance to the ground in 3.0 seconds. How long will it take to fall the second half? Ignore air resistance.

(Hint: Figure out the height of the cliff first!)

Choose an Origin and Positive DirectionEasiest choice!

y0 = 0 ma = 9.8 m/s2y will be positive

How High is the Cliff?A rock is dropped off a cliff and falls the first half of the distance to the ground in 3.0 seconds. For the first half of the fall, we can use y = v0t + at2where y is half the height of the cliff and t = 3 s

This gives y = * (9.8 m/s2) * (9 s2) = 44.1 m

Therefore, the cliff is 88.2 m high.Initial: Top of cliff (v0 = 0) Final: Halfway down

How long does it take to fall the second half?Now, we need to use y = v0t + at2, but with different initial and final conditions.

Initial: Halfway down (the rock will have a nonzero v0)Final: Just about to hit the ground.

y = 44.1 m (second half is just as far as first half)To find the rocks velocity halfway down, we must usevf = v0 + at, and we get vhalfway = 9.8 m/s2 * 3s = 29.4 m/s

Finishing the joby = 44.1 mv0 = 29.4 m/sa = 9.8 m/s2

y = v0t + at2

44.1 = (29.4)t + 4.9 t2

Sometimes (semi-rarely), you will need to use the quadratic equation to calculate time when the object has an initial position, initial velocity and a nonzero acceleration.

One of your homework problems uses it; make sure you also know the quadratic equation by heart.

Remember to check if a, b or c are negative.

44.1 = (29.4)t + 4.9 t2

Rearrange to get all terms on one side4.9 t2 + 29.4 t 44.1 = 0a = 4.9b =29.4c = -44.1

Using quadratic, we gett = -7.24 s or 1.24 s

The first half takes 3 secondsThe second half takes only 1.24 seconds!

Why?

But WAIT!!!!!!Theres an easier, non-quadratic-ier way to solve it!Once you find that the cliff is 88.2 m tall, and you know that the first half of the drop takes 3 secondsFind the total time of the whole drop, and subtract 3 seconds!This should get you the same result try it!

Tips for Solving Kinematics ProblemsRead critically (dropped, comes to stop, highest point reached)Direction of a vector determines its sign.Record all of the given informationMemorize the big three!You will not have a calculator on the AP Exam multiple choice, so if the calculations become too hard, go back and try a different approach

Final TipIf you encounter a toughie, try to step back, breathe and look at the scenario as a whole. There may be a shortcut or technique that makes the problem easier and saves you time!

This cant really be put into a series of steps, you just need to practice by studying hard and spending time on your homework

Meet the Position vs Time Graph

Interpreting x vs t graphsThe slope of the tangent line represents the instantaneous velocity of the object at anyone instant in time

The slope of a secant line represents the average velocity of the object between two moments in time.vav = x/t

Example from 2008 ExamThe figure to the left shows the position vs. time graphs for two objects, A and B, that are moving along the same axis. The two objects have the same velocity

(A) at t = 0 s.(B) at t = 2 s.(C) at t = 3 s.(D) at t = 4 s.(E) never

Example from 2008 ExamThe figure to the left shows the position vs. time graphs for two objects, A and B, that are moving along the same axis. The two objects have the same velocity

(A) at t = 0 s.(B) at t = 2 s.(C) at t = 3 s.(D) at t = 4 s.(E) neverv

Example from 2009 ExamThe graph below represents position vs. time for a sprinter at the start of a race. Her average speed during the interval between 1 second and 2 seconds is most nearly:(A) 2 m/s(B) 4 m/s(C) 5 m/s(D) 6 m/s(E) 8 m/s

Example from 2009 ExamThe graph below represents position vs. time for a sprinter at the start of a race. Her average speed during the interval between 1 second and 2 seconds is most nearly:(A) 2 m/s(B) 4 m/s(C) 5 m/s(D) 6 m/s(E) 8 m/s

However, x vs t graphs also have a SECRET PROPERTY hidden within them...but Im not going to tell you until tomorrow.

JUSTKIDDINGHEREWEGOThe concavity of the x vs t graph tells you the acceleration of the object!