Introduction to Motion Motion is Relative Straight Linear Falling Curved Circular/ Angular/ Rotational Projectile

Linear Motion Topics Distance vs. Displacement, Speed vs. Velocity, Acceleration, Free-fall, Average vs. Instantaneous quantities, Motion diagrams, Motion graphs, Kinematic formulas.

Distance vs. Displacement Tells “how far” an object is from a given reference point or “how far” it has traveled in a given time. Is a scalar quantity which means it has magnitude only Is measured in length units such as meters (m), centimeters (cm), feet (ft), inches (in), miles, etc. Displacement Tells “how far” and “which way” an object moves. Is distance in a given direction Is a vector quantity because it includes magnitude and direction The magnitude is measured in the same units as distance, direction is usually measured as an angle. “d” is the distance from A to B and the arrow indicates the direction of the displacement (from A to B). “d” is the distance from point A to point B or from B to A d d A A B B

Distance vs. Displacement Distance : 2+5+5+1+1+2 =16 m 5m 2m Displacement: 3m 5m start 45° @ 45° south of east 3m 1m N Displacement 1m W E 2m end S

Speed vs. Velocity Ex: speed = 60 mph Ex: velocity =60 mph, N Speed Is the rate of change of position of an object Tells “how fast” an object is moving Is a scalar quantity because it only includes magnitude (size). Measured in units of distance over time such as mi/hr, km/hr, or m/s Velocity Is speed in a given direction Tells “how fast” and “which way” Is a vector quantity because it includes magnitude (speed) and direction. Magnitude is measured in same units as speed. Ex: speed = 60 mph Ex: velocity =60 mph, N

Speed and Velocity Formulas Average speed = distance or v = d time t Average velocity = displacement or v =  d change in time  t

Speed & Velocity Problems The speedometer in every car has an odometer that records the distance traveled. If the odometer reads zero at the beginning of a trip and 35km a half hour later, what is the average speed? If a cheetah can maintain a constant speed of 25m/s, it will cover 25 m every sec. At this rate, how far will it travel in 10 sec? 1 min? The speedometer of a car moving northward reads 60mph. It passes another car that travels southward at 60 mph. Do both cars have the same speed? Do they have the same velocity? Mr. Steinke drove 150 miles north at 70 mph. After a stop of 15min, he continued north at 65 mph for 75 more miles. What was the total time of his trip? What was his average velocity?

Average vs. Instantaneous velocity Average velocity is the calculated by dividing total distance traveled by total time Constant velocity has the same value as average velocity Instantaneous Instantaneous velocity is the velocity that an object has at a specific instant in time. It is NOT the velocity over a time interval. Initial velocity and final velocity are examples of instantaneous velocities.

Acceleration The rate of change of velocity How fast the speed changes Measured in meters per second per second (m/s/s) or meters per second squared ( ) A vector quantity (requires both magnitude and direction) An object may accelerate (or decelerate) in three ways: Speed up Slow down Change direction If acceleration is… …positive then it is in the same direction as the velocity causing velocity to increase. …negative then it is in the opposite direction of the velocity causing velocity to decrease (deceleration)

Free-fall An object under the influence of only gravity is said to be in “free-fall” We assume no air resistance for any free-fall problems…thus, the “only gravity” stipulation. All objects fall at a constant acceleration… (9.81 m/s2 downward near earth’s surface)…regardless of mass. (For quick calculations and estimates we can round to 10 m/s/s) This means that a falling object will gain 9.81 (or 10) m/s of speed every second that it falls. Also…an object that is thrown up will slow down or lose 9.81 (or 10) m/s every second of its upward motion. The velocity of the object at the tip-top of the path is zero, the acceleration is “g” the acceleration due to gravity.

Motion Graphs: distance-time faster Distance Time Δt Acceleration (increasing slope) slower Time The slope of the line on a distance- time graph tells us about the speed. Steeper slope means faster speed Straight line means constant speed For non-constant motion, the instantaneous speed can be calculated by finding the slope of the tangent line at that point. Deceleration (decreasing slope) Distance Time

Motion Graphs: velocity-time The slope of the line on a velocity- time graph tells us about the acceleration. Steeper slope  higher acceleration Straight line means constant acceleration For non-constant motion, the instantaneous acceleration can be calculated by finding the slope of the tangent line at that point. Changing accelerations velocity Time

Motion Graphs: velocity-time 8 m/s Velocity Area The seconds cancel out and the units of the area are meters, therefore area is a displacement. Time 10 sec The area under the curve of a velocity-time graph for a particular time interval gives the displacement of the object during that time interval. Base = time Height = total velocity so another way of calculating displacement is d = ½ (time)(Vi + Vf)

Interpreting graphs (constant motion) acceleration velocity distance time time time Slope from this gives you this Slope from this gives you this The slope of the pink line is a lower positive constant value. The slope of the blue line is a greater positive constant value The slope of the green line is a negative constant value. A lower positive constant slope means a slower positive constant velocity (horizontal line). A greater positive constant slope means a faster positive constant velocity (horizontal line). A negative constant slope means a negative constant velocity (horizontal line). The slope of a horizontal line is zero, so the acceleration for all three cases is zero. All three motions (pink, blue, and green) are graphed along the time axis at acceleration = 0.

Motion Graphs: acceleration-time Note: The slope of a horizontal line is zero. The slope of a vertical line is undefined. Positive constant acceleration Zero acceleration (constant velocity) acceleration Time Negative constant acceleration Horizontal line means constant acceleration We will not do any calculations with changing accelerations, we assume all accelerations to be constant.

Interpreting graphs (accelerating motion) acceleration velocity distance time time time Slope from this gives you this Slope from this gives you this The slope of the red line starts at a fairly low (flat) value and is increasing (gets steeper) steadily over time. The slope of the green line starts at a higher value (it’s steeper) and decreases (gets flatter) over time. Increasing slope indicates an increasing velocity over time. Decreasing slope indicates a decreasing velocity over time. We will assume the changes are taking place at a steady rate. Since the slope of the velocity graph is a positive constant value we know that the acceleration is positive and constant, therefore a horizontal line above the axis on the acceleration graph. The slope on the velocity graph is negative and constant so the acceleration is a constant negative value, therefore a horizontal line below the axis.

Motion Diagrams A series of pictures illustrating the motion of the object including displacement, velocity, and acceleration. The rules: 4 images at equal time intervals…pay attention to spacing Include and label velocity vectors with relative lengths to represent relative speed on each image Include and label a single acceleration vector Example: Car moving to the right at constant velocity v v v v a=0 The term “constant velocity” means that the car will cover equal distances in equal time intervals…thus equal spacing. Also all velocity vectors are the same length indicating the same speed. If there is no change in speed, there is no (zero) acceleration.

Kinematics (motion) Formulas Calculating average values Calculating instantaneous values

Word clues to numbers for problem solving “free-fall”  acceleration due to gravity a=9.81m/s2, down “at rest”  not moving v=0 “dropped”  starts at rest and free-fall vi=0 and a=9.81m/s2, down “constant velocity”  no acceleration a=0 “stops”  final velocity is zero vf=0 These are the most common but be on the lookout for more.