Kinematics in Two Dimensions AP Physics 1

Cartesian Coordinates When we describe motion, we commonly use the Cartesian plane in order to identify an object’s position This is simply the x-y plane that you are familiar with from math class

Cartesian Coordinates When considering an object in Cartesian Coordinates, it is important to determine a reference (zero) point This is often where the object starts but can be an point that is convenient Regardless of the reference point, all calculations will give the same result

Vectors and Scalars Scalars –Most measurements you have used to this point are scalars –This means that they have a magnitude (size) –They include measurements such as mass, energy, distance, speed and time Vectors –Many measurements in Physics are vectors –In addition to a magnitude they also have a direction –Velocity, displacement, momentum and acceleration are all vector quantities

Position Vectors A position vector is simply a vector (arrow) that connects the reference point of a coordinate system to an object Reference Point Position Vector

Displacement Displacement is a vector quantity that measures the change in an object’s initial and final position

Time and Time Intervals In physics, we will often start timing when something occurs (this provides a zero in time) We may also consider a time interval which is symbolized as Δt

Velocity Velocity is a vector quantity that is the rate of change of position; it is calculated as: If we remove the directional information from the velocity, we are left with speed:

Position and time data can be analyzed using multiple representations: motion diagrams Vectors Graphs Equations Motion diagrams are a series of ‘dots’, numbered in succession and positioned to indicate direction Time interval between each dot is equal As an object’s speed increases, the dots on its motion diagram increase in separation As an object’s speed decreases, the dots decrease in separation

Examples of motion diagrams: Situation: A skateboarder rolling down the sidewalk at constant speed. A constant distance between the positions of the moving skateboarder shows that the object is moving with constant speed.

Examples of motion diagrams: Situation: A car stopping for a stop sign. A decreasing distance between the positions of the moving car shows that the object is slowing down.

Examples of motion diagrams: Situation: A sprinter starting a race. An increasing distance between the positions of the moving runner shows that the object is speeding up.

Examples of motion diagrams: Situation: A free throw in a basketball game. A more complicated motion (projectile motion) shows both slowing down (as the ball rises) and speeding up (as the ball falls).

Motion diagrams develop operational definitions for different motions, i.e. constant speed, slowing down, speeding up Operational definitions are those defined in terms of particular procedure or operation performed by an observer. Assume for now that motion is translational  along a path or trajectory An object is considered a particle, a mass at a single point in space Particles have no shape, size or distinction between front and back or top and bottom

Constant, Average and Instantaneous Velocity

Constant Velocity If an object is traveling at a constant velocity, a position time graph will result in a straight line (constant function) This is referred to as uniform or non- accelerated motion

Average Velocity It is rare that an object will travel at the same velocity throughout its trip so it is often useful to consider the average velocity The average velocity is taken between two points and is determined as the slope of a line connecting those two points

Instantaneous Velocity The instantaneous velocity is the velocity at one specific instant in time This is determined by drawing a tangent line to that point on the graph and determining the slope of the tangent line

Instantaneous Velocity Calculate the slope of the tangent line to find instantaneous velocity!

Acceleration

similar to how velocity is the rate of change of position w.r.t. time  determined by the slope of a line on a position-time graph acceleration is the rate of change of velocity w.r.t. time  the slope of a line on a velocity-time graph position time, velocity time and acceleration time graphs for a given situation are linked together

Acceleration

Examples of motion diagrams with position vectors: An object is at constant or uniform speed if its displacement vectors are the same length.

Examples of motion diagrams with position vectors: An object is slowing down if its displacement vectors are decreasing in length.

Examples of motion diagrams with position vectors: An object is speeding up if its displacement vectors are increasing in length.

Examples of motion diagrams with velocity and acceleration vectors: For constant velocity, vectors are represented by the zero vector,, or a dot (no arrow). Therefore, the acceleration vectors,, represented by the zero vector,, or a dot (no arrow). This is no acceleration or constant velocity. The operational definition is the separation of position on a motion diagram remains constant in equal time intervals.

Examples of motion diagrams with velocity and acceleration vectors: For an object slowing down at a constant rate, the vectors are the same and point in the opposite direction to motion. Therefore, the acceleration vectors,, are the same length but point in the opposite direction to motion. This is constant negative acceleration or slowing down in a positive direction. The operational definition of constant acceleration in this situation is the separation of position on a motion diagram decreases by the same amount in equal time intervals.

Examples of motion diagrams with velocity and acceleration vectors: For an object speeding up at a constant rate, the vectors are the same and point in the same direction as motion. Therefore, the acceleration vectors,, are the same length and point in the same direction as motion. This is constant positive acceleration or speeding up in a positive direction. The operational definition of constant acceleration in this situation is the separation of position on a motion diagram increases by the same amount in equal time intervals.

For motion along a line: An object is speeding up if and only if v and a point in the same direction. An object is slowing down if and only if v and a point in the opposite direction. An object’s velocity is constant if and only if a = 0.

A positive or negative acceleration DOES NOT indicate that an object is speeding up or slowing down. A positive acceleration can indicate a slowing down of an object in a negative direction OR a speeding up in a positive direction. Conversely, a negative acceleration can indicate a speeding up of an object in a negative direction OR a slowing down in a positive direction.

Acceleration Acceleration is a vector quantity the direction of both the velocity and acceleration is crucial to understanding the situation – –Positive velocity with positive acceleration (faster to the right/up) – –Positive velocity with negative acceleration (slower to the right/up) – –Negative velocity with positive acceleration (slower to the left/down) – –Negative velocity with negative acceleration (faster to the left/down)

Graphs are not pictures, but drawing pictures or pictorial representations that contain important information about a kinematics situation can provide a greater understanding of the motion. The steps to drawing a pictorial representation are: 1.Draw a motion diagram. 2.Establish coordinate system. 3.Sketch the situation. 4.Define symbols. 5.List knowns and unknowns. 6.Identify desired unknown. Pictorial Representations

1.List known and unknown values and what value one wishes to find. 2.Draw a pictorial representation. 3.Draw a motion diagram and graphical representation (if appropriate). 4.Develop a mathematical representation with formulae using the variables and values in the pictorial representation. Solve. 5.Assess the result. Is the answer reasonable? Check for appropriate units and significant digits. Problem-Solving Steps in Kinematics

Equations involving Constant Acceleration & Working with Kinematics Graphs

Kinematics Equations for Constant Acceleration

Sample Problem If a rocket with an initial velocity of 8.0 m/s at t = 0 s accelerates at a rate of 10.0 m/s 2 for 2.0 s, what is its final velocity at t = 2.0 s ?

Kinematics Equations for Constant Acceleration

Sample Problem What is the displacement of a bullet train as it is accelerated uniformly from +15 m/s to +35 m/s in a 25 s time interval?

Kinematics Equations for Constant Acceleration N.B.: If an object starts from rest, then v i = 0 m/s and d = ½ at 2 (i.e. this d-t graph looks like a parabola)

Sample Problem A car starting from rest accelerates uniformly at +7.2 m/s for 8.0 s. How far does the car move?

Kinematics Equations for Constant Acceleration Note: this equation does not involve time !

Sample Problem An airplane must reach 75 m/s for take-off. If the runway is 0.5 km long, what must the constant acceleration be?

Acceleration due to Gravity "g" is a vector quantity -g= m/s 2 (an average value across Earth) N.B.: neglect air resistance g can be substituted in equations for constant acceleration previously in notes

Sample Problem A 3.0-kg stone is dropped for a height of 5.0 m. How long does it take to reach the ground? What is its velocity at the moment it hits the ground?

Equations of Motion