Kinematics Uniform Motion

Clarification: Vector symbol Technically the variable for speed is v and the variable for velocity is v . In homework and on tests you do not need to be so formal. You can leave off the arrow over the v when it is velocity. In fact, I will often do this when solving problems in class and in these presentations. In addition, vector quantities are often converted into scalar quantities in order to solve certain mathematical processes. However, if the arrow is omitted it is still important for you to know when you are working with a scalar or a vector quantity.

Clarification: Final and initial conditions Initial Conditions are often annotated with an i subscript Initial time = ti Initial position = xi Initial velocity = vi Earlier we saw that we could spatially orient a problem by setting the origin of the coordinate system at the center of the object at the instant that the problem begins. To make elapsed time easier to calculate we can temporally orient a problem by setting initial time equal to zero ti = 0 . An alternate way to write initial conditions is with a zero subscript, which represents that condition when initial time was equal to zero. Initial time = t0 Initial position = x0 Initial velocity = v0 The subscript 0 = naught (Naught means nothing) Initial conditions for these variables can be referred to as: t initial, t zero, or t naught x initial, x zero, or x naught v initial, v zero, or v naught

Clarification: Final and initial conditions Final Conditions can be annotated with an f subscript Final time = tf Final x position = xf Final velocity = vf If we opt to use the zero subscript for initial conditions, then the f subscript for final conditions becomes unnecessary. Final time = t Final x position = x Final velocity = v You may see either convention employed in the text AP Central seems to favor the zero subscript for initial conditions and the no subscript for final conditions. As a result, this will be the convention I will use most frequently.

Clarification: Change (  ) Delta  symbolizes change Example t = change in time x = change in x axis position Change is always mathematically solved the same way Final conditions minus initial conditions For t this would be time final ( tf = t ) minus time initial ( ti = t0 ) t = tf  ti or t = t  t0 For x this would be time final ( xf = x ) minus time initial ( xi = x0 ) x = xf  xi or x = x  x0

Reality Check: Elapsed Time and Displacement In the previous slides we saw that t = t  t0 and that we usually start problems when t0 = 0 . Substitute t0 = 0 into the equation t = t  t0 . t = t  0 t = t Under these conditions the elapsed time equals the final time. We also saw that x = x  x0 and that we usually start problems at the origin x0 = 0 . Substitute x0 = 0 into the equation x = x  x0 . x = x  0 x = x Under these specific, but frequently encountered, conditions the magnitude of displacement equals the value of the final position. CAREFUL: While mathematically equal these letters represent different quantities. Many people simplify the equation by deleting the  symbols, and AP Central does not seem to care if you do, as long as it is clear that you understand which variable you are solving for. Dropping the  symbol is not wise in every case, and including it in some problems can be advantageous.

Objectives Define, compare, and contrast speed and velocity Describe and depict uniform motion of an object using written sentences a motion diagram a data table a position versus time graph and a velocity versus time graph an equation Determine the magnitude and direction of displacement and velocity from both a position-time graph and a velocity-time graph Solve problems involving uniform motion

Speed versus Velocity Speed A scalar quantity Magnitude only Magnitudes of scalars can be positive, negative, or zero Speed is the rate that distance (d) is traveled Rates in kinematics: Divide the stated variable (distance) by elapsed time

Speed versus Velocity Velocity A vector quantity Magnitude and direction Magnitudes of vectors are only positive or zero, and NEVER negative However, direction along an axis can be specified as negative Example: v = 50 m/s,  x direction Velocity is the rate of displacement (x) NOTE: A vector can be changed into a scalar quantity, for mathematical purposes, by moving the sign on direction in front of the magnitude. The vector v = 50 m/s,  x can be changed into the scalar vx = 50 m/s in specific mathematical applications to be explained soon.

Depicting Uniform Motion 5 10 15 20 25 30 35 40 5 m t (s) x (m) Δx (m) v (m/s) t0 = 0 x0 = 0 v0 = 5 1 5 2 10 3 15 4 20 25 6 30 7 35 8 40 Motion can be represented in several ways. A few are shown here. As a motion diagram In a table As a position versus time graph As a velocity versus time graph 2 4 6 10 8 40 30 20 t (s) x (m) 8 6 4 2 10 t (s) v (m/s)

Depicting Uniform Motion 5 10 15 20 25 30 35 40 5 m t (s) x (m) Δx (m) v (m/s) t0 = 0 x0 = 0 v0 = 5 1 5 2 10 3 15 4 20 25 6 30 7 35 8 40 One important aspects of graphs is their slopes. Examine the slope of position versus time. The rise is displacement and the run is elapsed time. The slope of position versus time is velocity. 2 4 6 10 8 40 30 20 x (m) v (m/s) t (s) t (s)

Depicting Uniform Motion 5 10 15 20 25 30 35 40 5 m t (s) x (m) Δx (m) v (m/s) t0 = 0 x0 = 0 v0 = 5 1 5 2 10 3 15 4 20 25 6 30 7 35 8 40 Another important aspects of graphs is the area bounded by the function and the x-axis. Examine the area of velocity versus time. The height is velocity and the base is elapsed time. The area of velocity versus time is displacement. NOTE: Area can be a bit tricky as it solves for change in position rather than position. This distinction will be detailed in class. 2 4 6 10 8 x (m) v (m/s) 40 30 20 t (s)

Other Graphs of Uniform Motion Object is on the right of origin (+x), and it is moving to the right (+v) Object is on the right of origin (+x), and it is moving to the left (v) Object is on the left of origin (x), and it is moving to the left (v) Object is on the left of origin (x), and it is moving to the right (+v) x t v