PH 201 Dr. Cecilia Vogel Lecture 3

REVIEW  Motion in 1-D  instantaneous velocity and speed  acceleration OUTLINE  Graphs  Constant acceleration  x vs t, v vs t, v vs x  Vectors  notation  magnitude and direction

Sign of Acceleration  Mathematically  If (signed) velocity increases, a is +  If (signed) velocity decreases, a is -  Memorize  If velocity and acceleration are in same direction, object will speed up  If velocity and acceleration are in opposite directions, object will slow down  Physical intuition  positive acceleration produced by push or pull in + direction  negative acceleration produced by push or pull in - direction

Position, Velocity, Acceleration  Velocity is  slope of tangent line on an x vs t graph  limit of  x/  t as  t goes to zero  the derivative of x with respect to time  dx/dt  Similarly acceleration is  slope of tangent line on a v vs t graph  limit of  v/  t as  t goes to zero  the derivative of v with respect to time  dv/dt  If you have position as a function of time, x(t)  can take derivative to find v(t)  take derivative again to find a(t)

Derivatives of Polynomials  The derivative with respect to time of a power of t, if C is a constant:  Special case, if the power is zero:  The derivative of a sum is sum of derivatives:  ex

Example  ex  The acceleration at t=0 is -6 m/s2, and at t=3 is 90 m/s2.  The average acceleration between t=0 and t=3 is 39 m/s2

Special Case: Constant Velocity  Acceleration is zero  Graph of x vs. t is linear  slope is constant  Average velocity is equal to the constant velocity value, v if initial time is zero, and we drop subscript on final variables. becomes

Special Case: Constant Acceleration  If object’s acceleration has a constant value, a,  then its velocity changes at a constant rate:  And its position changes quadratically with time:

Position with Constant Acceleration  Slope of the position graph (velocity) is constantly changing  quadratic function of time.

Example A little red wagon is rolling in the positive direction with an initial speed of 5.0 m/s. A child grabs the handle and pulls, giving it a constant acceleration of 1.1 m/s 2 opposite its initial motion. Let the time the child begins to pull be t=0, and take the position of the wagon at that time to be x=0. a)How fast will the wagon be going after 1.0 s of pulling? b)Where will the wagon be then? c)At what time will the wagon come to a stop (for an instant)?

What if…?  What if I asked “where will the wagon be when it is going -1.0 m/s?”  You could:  find the time that v= -1.0 m/s  find the position at that time.

What if…?  Let’s find a generalization of that:  Where will object be when it’s velocity is v, given a known initial position, velocity, and constant acceleration ? simplify:

Derivatives and Constant Acceleration Yeah – consistency!