PH 201

1
PH 201

• Dr. Cecilia Vogel
• Lecture 2

2
REVIEW

• Motion in 1-D
• position
• displacement
• average velocity
• Special case
• Constant velocity

OUTLINE

• Motion in 1-D
• velocity and speed
• acceleration
• velocity and acceleration from graphs

3
Summary

• Positive velocity
• Moving forward
• Value of position, x, increasing
• Slope of x vs. t is positive
• Negative velocity
• Moving backward
• Value of position, x, decreasing
• Slope of x vs. t is negative
• Zero velocity
• Stopped (maybe just for an instant)
• Occurs any time motion changes direction
• x vs. t graph is horizontal (maybe just for an
  instant)

4
Average Speed

• Average speed is the distance traveled divided by
  the time interval.

• Compare to
• Average velocity is displacement divided by time
  interval
• Small wording difference, big difference in
  results

Ex Object goes 5 cm in the negative direction
then two cm in the positive direction, all in one
second. vave -3 cm/s, save7 cm/s
Ex Object goes 5 cm in the negative direction in
one second. vave -5 cm/s, save5 cm/s
5
Ave Speed Example

• Example
• Sharrona drives to her friends house along a
  straight road at a constant 35 mph. She forgot
  something, so returns home, traveling only 25
  mph.
• Find her average velocity and average speed.
• Problem-solving hints.

6
Average Velocity on a Graph

• On a graph of position vs. time, average velocity
  is the slope of a line connecting two points on
  the graph.

7
Velocity
Average velocity, given by slope of straight
line, doesnt describe everything that happens
during the time interval. (Instantaneous)
velocity describes what happens at a given
instant at a single point on the graph.
8
Instantaneous Velocity
If you take a smaller time interval, you more
accurately reflect whats happening at that
instant.
9
Velocity on Graph

• On a graph of position vs. time, (instantaneous)
  velocity is the slope of a line tangent to the
  curve at that instant.

10
Instantaneous Speed

• Instantaneous speed is just the magnitude of the
  instantaneous velocity.
• Always positive.
• Note it is also the limit as Dt approaches zero
  of the average speed.
• As time interval goes to zero, theres no time to
  change direction, so speed v

11
Acceleration

• Average acceleration is
• Change in velocity over the time interval

• Objects acceleration is positive if
• vf gt vi
• velocity is increasing
• Remember that from -50 to -40 is an increase
• Speed NOT necessarily increasing!!

12
Direction of Acceleration

• Three ways to find acceleration direction
• Is velocity (not speed!) increasing?
• if so, accel is
• dont forget -10 gt -50
• What direction do you feel tossed when you
  accelerate that way?
• imagine a car
• acceleration is OPPOSITE that
• What way would you have to push/pull to cause
  that acceleration?
• acceleration is in that direction

13
Examples Way 1

• Velocity positive, speed increasing
• ex vi 30 mph, vf 60 mph
• velocity is increasing, a is _____
• Velocity positive, speed decreasing
• ex vi 30 mph, vf 20 mph
• velocity is decreasing, a is __-___

14
Way 1, contd

• Velocity negative, speed increasing
• ex vi -30 mph, vf -60 mph
• velocity is decreasing, so a is _-__
• (speeding up, but negative accel!!!)
• Velocity negative, speed decreasing
• ex vi -30 mph, vf -20 mph
• velocity is __creasing, so a is __

15
Way 2

• Your body is an accelerometer, you feel pushed
  opposite the acceleration.
• Velocity positive, speeding up
• feel pushed back
• opposite that, a is back or -
• Velocity positive, slowing down
• feel pushed _____
• opposite that, a is ____

16
Way 2 contd

• Velocity negative, speeding up
• feel pushed forward
• opposite that, a is back or -
• Velocity negative, slowing down
• feel pushed ___
• opposite that, a is ___

17
Sign of Acceleration

• Acceleration requires a force. How would you
  have to push something to change its velocity?
• Objects velocity is positive, if you want it to
  speed up
• must push forward,
• Push forward -gt a is forward or
• Objects velocity is positive, if you want it to
  slow down
• must push backward,
• push back -gt a is ___

18
Sign of Acceleration

• Objects velocity is negative, if you want it to
  slow down
• push forward,
• so a is forward or
• Objects velocity is negative, if you want it to
  speed up
• push backward,
• so a is ___

19
Acceleration on a Graph

• On a graph of velocity vs time, acceleration is

• However, on a graph of position vs. t,
• velocity is the slope,
• what is accel?

20
Acceleration on a Graph

• In this region, v (slope) is decreasing
• Acceleration, then, is negative

21
Acceleration on a Graph

• In this region, v (slope) is
• Acceleration, then, is

22
Acceleration on a Graph

• In this whole region, slope is decreasing, the
  curve is concave down
• Acceleration, then, is negative
• Generally concave down is negative accel

23
Acceleration on a Graph

• In this region of time, slope is increasing, the
  curve is concave up
• Acceleration, then, is positive
• Generally concave up is positive accel

24
Summary

• Positive acceleration
• Moving forward and
• Or moving backward and
• Or stopped and starting forward
• _______ curvature on x vs t graph U
• Negative acceleration
• Moving backward and
• Or moving forward and
• Or stopped and starting backward
• ________ curvature on x vs t graph n