Transcript Chapter 2 - Motion in One Dimension
Chapter 2 - Motion in One Dimension
2.1 - Displacement and Velocity
Motion
• One-dimensional motion is the simplest form of motion • e.g. a train on a straight track (forward or backward) • In this chapter we will consider only one-dimensional motion
Motion
• The first step in analyzing motion is to choose a frame of reference.
• If an object is at rest, its position does not change with respect to a frame of reference.
• When we choose this frame of reference, we need to remember to remain consistent.
Displacement
Displacement is the change in position of an object.
It is the length of the straight line drawn from its initial position to its final position
x
x f
x i
Displacement = change in position = final position - initial position
Displacement
• Displacement is not always equal to the distance traveled
Displacement
• Displacement can be positive or negative
Average Velocity
• Knowing the distance traveled or the displacement doesn’t tell completely describe the motion of an object • Knowing the speed is important for evaluating motion • Average velocity is the total displacement divided by the time interval during which the displacement occurred
v avg
x
t
x f t f
x i
t i
Average Velocity
• The Average Velocity can be positive or negative • Equal to the constant velocity needed to cover the given displacement in a given time interval • It is NOT the average of the starting and ending velocities – Example: A car travels from city A to city B (100 km). If the first half of the distance is driven at 50 km/h and the second half is driven at 100 km/h, what is the average velocity of the car?
Practice
• Page 44, Practice 2A
Average Speed
• Velocity is not the same as speed – Velocity has both magnitude AND direction – Speed has only a magnitude average speed = distance traveled time of travel
Graphical Interpretation of Velocity
• In a position-time graph, we can determine the v avg a straight line between two points on the graph by drawing
Graphical Interpretation of Velocity
slope rise run change in vertical coordinates change in horizontal coordinates
v avg
x
t
x f
x i t f
t i
Instantaneous Velocity
• The velocity of an object at some instant
Chapter 2
Section 2.2 - Acceleration
Acceleration
• Most objects in motion don’t move with a constant velocity • Acceleration measures the rate of change in velocity in a given time interval
a avg
v
t
v f t f
v i
t i
change in velocity average acceleration = time required for change (
m
/
s
)
s m s
1
s
m s
2
Problem
• Find the acceleration of an amusement park ride that falls from rest to a speed of 28m/s in 3.0s.
• 9.3 m/s 2
Acceleration
• Acceleration has both magnitude and direction • If an object (i.e. a car) is moving in the positive direction and is speeding up, the acceleration is positive • If an object is moving in the positive direction and is slowing down, the acceleration is negative • If an object (i.e. a car) is moving in the negative direction and is speeding up, the acceleration is negative • If an object is moving in the negative direction and is slowing down, the acceleration is positive
Constant acceleration
• The slope of a velocity-time graph gives the acceleration • When acceleration is constant, the velocity is increased by the same amount during each time interval.
• The displacement for each time interval increases by the same amount
Displacement
• Depends on: – Initial velocity – Acceleration – Time When acceleration is constant, V avg is the average of V i and V f
Displacement
• When acceleration is constant:
v avg
v i
v f
2
x
t
v avg
v i
v f
2
x
1 2 (
v i
v f
)
t
Displacement
• The area under the curve in a graph of velocity versus time equals the displacement during the time interval
Problem
• A bicyclist accelerates from 5.0 m/s to a velocity of 16 m/s in 8 s. Assuming uniform acceleration, what distance does the bicyclist travel during this time interval?
• 84m
Final Velocity
• Depends on: – Initial velocity – Acceleration – Time
a
v f t f
v i
t i
v f
t v i a
t
v f
v i v f
v i
a
t
Constant Acceleration
Displacement with constant uniform acceleration
x
1 2 (
v i
v f
)
t
x
1 2 (
v i
v i
a
t
)
t
x
1 2 ÍÍ ÍÍÍ Î È 2
v i
t
a
(
t
) 2 × Ý Þ
x
v i
t
1 2
a
(
t
) 2
Final velocity after any displacement
v f
2
v i
2 2
a
x
Remember that the square root may be either positive or negative
Problem
• An aircraft has a landing speed of 302 km/h. The landing area of an aircraft carrier is 195 m long. What is the minimum uniform acceleration required for a safe landing?
• -18.0 m/s 2
Summary
HW Assignment
• Page 49, Practice 2B, 1 and 4 • Page 53, Practice 2C, 3 and 4 • Page 55, Practice 2D, 2 - 4 • Page 58, Practice 2E, 2, 5, 6
Chapter 2
Section 2.3 - Falling Objects
Free Fall
• Motion of an object falling with a constant acceleration • In the absence of air resistance all objects dropped near the surface of a planet fall with the same constant acceleration • The symbol for free-fall acceleration is “g” • At the surface of the earth, the magnitude of g is approximately 9.81 m/s 2 .
Free Fall
• This acceleration is directed downwards – a = -g = -9.81 m/s 2 • What goes up, must come down – Fig 2-15, pg 61 • Initial velocity of +10.5 m/s, acceleration is -9.81 m/s 2 • After 1 s, v = +0.69 m/s, acceleration is -9.81m/s 2 • After 2 s, v = -9.12 m/s (directed downward), a = 9.81 m/s 2
Free Fall
10 5 0 -5 -10 -15 -20 0 0.5
1 1.5
Time (s) 2 2.5
3
Free Fall
• FREELY FALLING OBJECTS ALWAYS HAVE THE SAME DOWNWARD ACCELERATION
Problem
• A ball is thrown straight up into the air at an initial velocity of 25.0 m/s. Create a table showing the ball’s position, velocity, and acceleration each second for the first 5.00 s of its motion.
• Find the ball’s time, position, velocity, and acceleration at the top of its flight
HW Assignment
• Pages 72 - 75: 34, 38, 40, 41, 46, 50, 56