1-D Kinematics: Concepts, Equations, Tips and Tricks!

Get those neurons firing!

A bunny travels in a straight line at a speed of 60 miles per hour for 30 minutes, and then continues in the same direction at a speed of 20 miles per hour for one hour.

What is the average velocity of the bunny for the entire trip?

vavg = Δx/Δt• To keep everything organized and your brain

free of clutter, make a table that breaks up the journey into part 1, part 2, and total.

Part 1 Part 2 Total

ΔxΔt

vavg

60 mi/hr 20 mi/hr

0.5 hrs 1 hr

30 mi 20 mi

33.3 mi/hr

1.5 hrs

50 mi

Proportional reasoning - Give it a go!

A ball is dropped off of a cliff of height h. Its velocity just before hitting the ground is v. How far (in terms of h) will it have fallen by the time it reaches a velocity of v/4?

Yes, we will very often be working with strictly variables – it is really no different than using numbers, and will become less and less scary as you get better at it :)

vf2 = v0

2 + 2aΔx vf2 = v0

2 + 2aΔy or

“Dropped”: v0 = 0

vf2 = 2aΔy

Δy is the distance that the ball has fallen from the point of release (top of cliff)

When using proportional reasoning, always solve for (isolate) the quantity that you are trying to figure out! Δy = vf

2/(2a)Δy α vf

2

Whatever factor vf is multiplied (or divided) by, Δy will be multiplied (or divided) by the square of that factor!

(Footnote: Proportional reasoning does not apply to addition or subtraction)

Δy α vf2

A ball is dropped off of a cliff of height h. Its velocity just before hitting the ground is v. How far (in terms of h) will it have fallen by the time it reaches a velocity of v/4?Divide v by 4 Must divide Δy by 16

Answer: The ball will reach 1/4th of its top speed when it falls a distance of h/16!

Benefit of using variables rather than numbers?

We can now apply this to any cliff, ever. In only one quick step.

Key Concepts of Kinematics• Velocity is a vector that defines an object’s speed

and direction of motion.• Acceleration is a vector that defines the change of

an object’s velocity.– An object is only accelerating if its velocity is changing!

• There are a few ways that an object can accelerate– A change of speed (speeding up or slowing down)– A change of direction (turning)– A simultaneous change of both

Tricky but crucial questions.1) “Can an object have an acceleration of zero, but

a nonzero velocity?”

Common incorrect response: “No. If an object is not accelerating, it can’t be moving” :-/

Rebuttal: “Acceleration does not mean motion! It means the change in an object’s motion. If an object moves at a constant velocity (steady speed in a straight line), will have zero acceleration, but nonzero velocity.”

Tricky but crucial questions.2) “Can an object have a velocity of zero, but a

nonzero acceleration?”

Common incorrect response: “No. If an object has zero velocity, it can’t have an acceleration” :-P

Rebuttal: “An object can have an instantaneous velocity of zero, but still be accelerating! The peak of an object thrown straight upward is proof of this. Its velocity is always changing, and is only zero for one exact instant. Boom.”

v vs t graph for an object launched upward

Its velocity is constantly changing (therefore it is accelerating)At the peak, its velocity is still changing, and is zero for only one instant.

Fun fact: An instant is an infinitely small unit of time!

From 2010 AP Physics B Exam

A ball is thrown straight upwards. If air resistance is ignored, which of the following is true at the ball’s highest point?

I. The velocity is 0 m/s. II. The acceleration is 0 m/s2. III. The acceleration is downwards. 

(A) I only(B) II only(C) III only(D) I and II only(E) I and III only

Conceptual Whiteboard

Tricky but crucial questions.3) “What does the sign of acceleration mean?”

Common incorrect response: “Positive acceleration means speeding up and negative acceleration means slowing down” D-:

Rebuttal: “Sign tells the direction of an object’s acceleration. If its acceleration is in the same direction as its velocity, it is speeding up. If the acceleration is opposite to the object’s velocity, it is slowing down.”