Day 3 UNIT 1 Motion Graphs x t Lyzinski Physics

Any desired information will be given, but only if asked for 

Day #3. instantaneous velocity. d-t vs Day #3 * instantaneous velocity * d-t vs. x-t graphs (what’s the difference???)

Definition Dt0 Honors Only!!! Instantaneous Velocity (v) – the velocity of an object at a precise moment in time. v = lim(Dx/Dt) Dt0 Honors Only!!!

Just what is meant by “instantaneous” velocity? Dt Dt Dt Dt Finally, “in the limit” that the time interval is infinitely small (or approximately zero), we find the velocity at a single moment in time.  Hence the term “instantaneous velocity” Dt To find the average velocity between two points in time, we find the slope of the line connecting these two points, thus finding the change in position (rise) over the change in time (run). As the two points move closer together, we find the average velocity for a smaller time interval. As the two points move EVEN CLOSER together, we find the average velocity for an EVEN SMALLER time interval.

To find instantaneous velocities, we still use the concept of slope To find instantaneous velocities, we still use the concept of slope. But we find the slope OF THE TANGENT TO THE CURVE at the time in question Definition Tangent to a Curve – a line that intersects a given curve at exactly one point.

Good Tangents  Bad Tangents 

How to find the instantaneous velocity of a specific time interval from an x-t graph … x(m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the instantaneous velocity at t = 20 seconds? (24, 30) (15, 15) Draw the tangent to the curve at the point in question. Then, find the slope of the tangent. Slope = rise/run = 15 m / 9 s = 1.7 m/s (approx) YOU MUST SPECIFY WHICH POINTS YOU USED WHEN FINDING THE SLOPE!!!!

How to find the instantaneous velocity of a specific time interval from an x-t graph … x(m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the instantaneous velocity at t = 5? If the pt lies on a segment, find the slope of the segment. Slope = 5 m / 10 s = 0.5 m/s (0,5) (10,10) YOU MUST SPECIFY WHICH POINTS YOU USED WHEN FINDING THE SLOPE!!!!

How to find the instantaneous velocity of a specific time interval from an x-t graph … x(m) 10 20 30 40 50 t (s) 30 20 10 Example: What is the instantaneous velocity at t = 25 seconds? Draw the tangent to the curve at the point in question. Then, find the slope of the tangent. Slope = 0 (object at rest)

x-t graphs 2 1 3 x (m) x2 x1 x3 t (sec) t1 t2 t3 Slope of line segment Slope of line segment Slope of line segment

1 Open to in your GREEN packet (0,6) (33,2) (11,-20) (13.5,-20) Tangent to the curve has a slope of +22m / 22sec = 1 m/s Tangent to the curve has a slope of -26m / 13.5s = -1.93 m/s THEREFORE, v = -1.93 m/s and s = 1.93 m/s (approximately)

HONORS ONLY Given a function for position, like x(t) = 3t2 + 3t – 6, find the instantaneous velocity at a give time (like t = 2 sec) using your graphing calculator. Step #1) Type the equation into “y =“ as y = 3x2 + 3x - 6 Step #2) Use “2nd-Trace-6” to use the “dy/dx” function Step #3) Since “dy/dx” is really dx/dt, type in “2” and hit enter to get the instantaneous velocity at t = 2 sec. ANSWER: dx/dt = 15 m/s.

HOMEWORK Check out your Unit 1 Schedule … Day #3 We will “try” to follow it night by night. Short Quiz Tomorrow (20 minutes)