Unit 1 - Kinematics Vectors, Graphical and Analytical Analysis of Linear Motion, Free Fall, and Projectile Motion

A Definition of... Vector - a quantity that involves both magnitude and direction. Scalar - a quantity that does not involve direction. For example, 55 miles per hour is a scalar, while 55 miles per hour east is a vector. Most things in physics will be vectors. Common physics scalars: work, energy, mass, power, temperature, electric charge

Commutative law of addition The vector sum A + B means the vector A followed by B. Two vectors are equal when they have the same magnitude and the same direction. Example 1. Add the following two vectors. A B

Scalar multiplication A vector can be multiplied by a scalar (that is a number) and the result is a vector. The magnitude of kA = |k|(magnitude of A) The direction of kA = the same direction as A if k is + the opposite of A is k is - Example 2. Sketch the multiples of 2A, -A, 0.5A, and -3A. A

Vector subtraction To subtract one vector from another, A - B, simply form the (-1)B vector and add it to A Example 3. For the two vectors A and B find A-B. A B

Two-Dimensional Vectors Two-dimensional vectors are vectors that lie flat in a plane and can be written as a sum of a horizontal and vertical vector. Think of your vectors as lying in a coordinate system, where the x-coordinate is your horizontal component vector and the y-coordinate is your vertical component vector.

Two-Dimensional Vectors The magnitude of a vector can be computed by Pythagorean theorem. In the picture below, c is the resultant vector with x-component a and y-component b. The direction of a vector can be determined with any trigonometric function. Note: now b is x-component and a is y-compnent of vector c nn

Vector Equations **MEMORIZE** Ax = Acos(θ) Ay = Asin(θ)

PHAST PHYSICS REVIEW _ VECTORS!

Position Example 1. A rock is thrown straight up from the edge of a 30 m cliff, rising 10 m then falling all the way down to the base of the cliff. What are the rock’s total distance and displacement? Distance = 40 m Displacement = -30 m down

Position Example 2. An infant crawls 5 m east, 3 m north, then 1 m east. Find the magnitude and direction of the infant’s displacement. Displacement = 6.7 m 26.6°N of E

A Definition of... Average Speed - the ratio of the total distance traveled to the time required to cover that distance. Average Velocity - the ratio of displacement traveled to the time required to cover that distance. **BIG IDEA** Speed is the magnitude of Velocity...Velocity is Speed plus direction! Note: the magnitude of average velocity is not called average speed.

Average speed and velocity Example 1. Assume that a runner moves a total distance of 500 m once around a circular track from start to finish. If they complete this in 1 min and 18 sec, find their average speed and average velocity. Is it possible to move with constant speed but not constant velocity? Is it possible to move with constant velocity but not constant speed?

Acceleration defined... Average Acceleration - the change in velocity over a period of time **SPECIAL NOTE** Acceleration is a change in velocity, and remember velocity is a vector, so we are talking about a change in either magnitude or direction. Ergo, acceleration is possible when there is a change in the (1) speed, (2) direction, or (3) both speed and direction.

Acceleration equation Example 2. A car is traveling along a straight highway at a speed of 20 m/s. The driver steps on the gas pedal and, 3 seconds later, the car’s speed is 32 m/s. Find its average acceleration.

Straight-line motion (1D) Uniform, or constant, acceleration is the simplest type of motion to analyze. There are two possible directions of motion (positive and negative). We have seen five kinematic quantities: Δx = displacement (in meters) t = time (in seconds) vₒ = initial velocity (in meters per second) v = final velocity (in meters per second) a = acceleration (in meters per second²)

Kinematic equations x a t v Δx = displacement (in meters) t = time (in seconds) vₒ = initial velocity (in meters per second) v = final velocity (in meters per second) a = acceleration (in meters per second²) Kinematic equations They can be arranged into four kinematic equations: missing variables x a t v NOTE: You can only use these equations in a system with uniform/constant acceleration. If it says constant speed/velocity in the problem, then you will be using the average speed equation.

Kinematic equations How to solve: Example 3. A car that’s initially traveling at 10 m/s accelerates uniformly for 4 seconds at a rate of 2 m/s², in a straight line. How far does the car travel during this time? Given: vₒ = 10 m/s, t = 4 s, a = 2m/s² Unknown: x = ? Equation: Substitute & Solve: x = (10)(4) + (0.5)(2)(4)² = 56 m

Special cases Leaving units off of your answer will cost you points on the AP!!! Some problems will not give initial velocity. If you see “starting from rest”, “dropped”, “falling”, etc, then you can assume initial velocity is zero. We will get more into the dropping and falling cases later in free fall.

Kinematics with graphs First let’s brush up on coordinate geometry. Equation of a line is y = mx + b, where a is slope and b is y-intercept

Consider the following... What does this graph tell us? @ t=0 the object is at x=0 In first two seconds, its position changes from x=0 to x=3m. From t=2 to t=4 s, it stopped. From t=4 to t=9 s, it reversed direction, reaching x=0 again at t=7s, and continued reaching position x=-2m at t=9s. It then remained at rest for another second to finish at t=10s. Consider the following...

Consider the following... What does this graph tell us? We can also determine average velocity… The slope of a position-vs-time graph gives the velocity. What are the four slopes? What is the average speed? How is this different from finding average velocity? Consider the following...

Changing velocity on graphs??? Not all graphs are as easy and pretty as that last one. When velocity changes the “lines” become “curves”! Example 1. What is the average velocity for the following time intervals: t=0 to t=2s b. t=2 to t=5s c. t=0 to t=5s

Instantaneous velocity The instantaneous velocity is the velocity at a given moment in time. To find the instantaneous velocity at t=3s you will need to draw a tangent line on the curve at t=3s. Notice we can make two points at (2.5,10) and (5,40) on this line in order to get the slope, which will be our instantaneous velocity.

Velocity vs time graph What does this graph tell us? At t=0, the velocity is v=0. Over the first 10 min, its velocity increases to 60 m/min. At t=10min it stays at a constant speed of 60 m/min until t=15min. At t=15min it changes it begins to decrease its velocity, eventually becoming v=0 again at t=30min. The velocity then becomes negative at t=30min, reaching a velocity of - 40m/min at t=40min. At t=40min, the velocity begins to decrease, arriving at v=0 again at t=55min.

Velocity vs time graph What does this graph tell us? We can tell the average acceleration… The slope of a velocity vs time graph is the acceleration. We can also tell the displacement of an object… The area under the curve/line is the displacement. (Note: there are both positive and negative areas! Think of it as a scalar that can be both negative or positive) What are the four slopes? What is the total displacement?

Qualitative graphing All graphs have a few basic shapes, and being able to quickly identify the situation will help you on the AP.

Identifying from x vs t graphs How would the velocity and acceleration graphs look in each case?

To be cont ... free fall projectile motion (2D)