Motion in One Dimension
DISPLACEMENT AND VELOCITY
To describe motion, we must first specify ______. But to specify position, we need to relate it to ______.
For motion in one dimension, it is convenient to use the ______.
From this concept, we can create a number line.
Displacement of an object –
NOTE: displacement is not always equal to the total distance traveled.
where, x =
xf=
xi=
Example
: If a car moves from an initial position of 10 m to a final position of 80 m:
Example: If a car moves from an initial position of 80 m to a final position of 20m:
Note: the negative sign indicates the direction of displacement.
Example: Every morning you drive __ miles from home to NPHS and then come back home at night using the same route. What is the best statement describing your daily trip? (Assume you do not move far while at NPHS) displacement = ______and distance traveled = _____
VELOCITY –
• The SI unit for velocity is ______; however, a velocity can be given in other units such as:
• When calculating the velocity, be sure all your ______. If they don’t, you need to ______
Example: It takes ____ hours to drive to San Francisco from Newbury Park if driving at 65 mi/hr. How far is the trip?
Example: The United States is 4,300 km wide. How long (in ______) would it take to drive across the country if someone were to drive at a steady 22 m/s the whole way?
Speed vs. Velocity
· Speed is a ______meaning it is only a ______, direction doesn’t matter.
· Velocity is a ______meaning it has a number (______) and ______. Direction matters.
Speed = Velocity =
ACCELERATION –
Units – acceleration has units of velocity/time, or…
Example: A car is initially coasting up hill at ______. 4.75 seconds later, the car is now rolling backwards at 2.6 m/s down the same hill. What is the car’s acceleration during this time period?
Time Graphs
Both displacement and velocity can be graphed as a function of time
Example 1 Example 2
Example 3 Example 4
SPEED vs. VELOCITY
Both describe how fast the ______is changing with respect to ______.
Speed is a ______quantity. It indicates ______(magnitude), but not direction.
Velocity is a ______quantity. It indicates both ______(magnitude) and ______.
SLOPE REVIEW
slope =
+ slope: -slope: 0 slope: slope undefined:
VELOCITY
Velocity is represented by the ______of the curve on a displacement vs. time graph.
In this class, a positive (+) slope indicates a forward direction and a negative (-) slope indicates a backwards direction (return).
Slope of a position vs. time graph =
Example: Use the position graph to answer the following:
a. What is the object’s velocity from 10 – 15 seconds?
b. What is the object’s velocity from 15 – 25 seconds?
c. What is the object’s velocity from 0 – 40 seconds?
Example: The x(t) graph describes a 1-D motion of a train. What must be true about this motion?
Example: The x(t) graph displays motions of two trains A and B on parallel tracks. Which statement is true?
ACCELERATION
Motion can be described with a velocity vs. time graph.
For a velocity vs time graph:
Slope =
Example: Use the velocity graph to answer the following:
a. What is the object’s velocity from 4 – 7 seconds?
b. What is the object’s acceleration from 4 – 7 seconds?
c. What is the object’s acceleration from 2 – 4 seconds?
Example: A train travels at 5 miles per hour for 1 hour. What is its displacement after 1 hour?
Displacement can also be determined by finding the area under the curve (AUC) of a velocity vs. time graph.
Graphical analysis summary:
Displacement vs. Time graph:
-
Velocity vs. Time graph:
-
-
Acceleration vs. Time graph:
-
Motion Maps:
* Imagine a toy car traveling along a piece of paper and dropping a dot of ink at a given time interval (say 1 drop every second). It could produce a trail that looks like this:
* How long does it take the car to travel the length of the paper?
* Describe the car’s motion:
Example: Draw the motion map for the following complex motion: Object accelerates for 3 seconds. Then travels at a constant velocity for 2 seconds. Then decelerates for 3 seconds. Stops for 2 seconds. Then returns to the start in 4 seconds at a constant velocity.
VELOCITY WITH CONSTANT ACCELERATION
Find the AUC…
5 Parameters of Motion:
1. a =
2. Δx=
3. vf=
4. vi=
5. t=
To solve a constant acceleration problem, you must know, or be able to find, three of the five parameters. Then use the following equations to solve for the other two: variable missing:
vf = vi + aDt
Dx = ½(vi + vf)Dt
Dx = viDt + ½a(Dt)2
vf2 = vi2 + 2aDx
Example: A jet plane lands with a velocity of ____ m/sec and can slow down (-acceleration) at a maximum rate of –5.0 m/s2. Find (a) the time required for the plane to come to rest, and (b) the minimum size of the runway.
(a) (b)
Example: A train is traveling down a straight track at ___ m/sec when the engineer applies the brakes, resulting in an acceleration of –1m/sec2 as long as the train is in motion. How far does the train travel in the first ___ seconds after the breaks are applied?
FALLING OBJECTS
A body is said to be in ______ when the only force acting on it is gravity.
A body in free fall near the surface of the Earth will have a constant acceleration.
This acceleration due to gravity, g =
This implies that heavy objects fall at the same rate as light objects and in a vacuum they do.
Quick Quiz: An object is dropped from rest at t = 0s. It falls freely with constant acceleration of 9.8 m/s2, which implies that:
Quick Quiz: Ignoring air resistance, if you drop an object, it accelerates downward at 9.8m/s2. If instead you throw it down, what will be its acceleration after you release it?
Quick Quiz: You throw a ball straight up. At its highest point, what are the magnitudes of ball’s velocity and acceleration?
Example:
A ball is thrown upward from the top of a 35.0m building. The ball has an initial velocity of 6.5 m/s.
Diagram:
A.
Calculate how high the ball will rise.
B. Calculate how fast will the ball be going when it hits ground.
Free Fall Motion: Free Fall Motion – Symmetry: