Post on 20-Jan-2016
transcript
Welcome to Physics C Welcome to your first year as a “grown-up” (nearly!)
What are college physics classes like?
Homework for the AP student
Keeping organized: Lab Notebook & Notes Binder
Information CardBirthday
Email address & cell #
Something unique about yourself
Favorite Hobby
(Back of notecard)
Lesson
Average Speed, Velocity, Acceleration
Average Speed and Average Velocity
Average speed describes how fast a particle is moving. It is calculated by:
Average velocity describes how fast the displacement is changing with respect to time:
always positivedistanceaverage speed
elapsed time
ave
xv
t
sign gives direction (the direction in which the object is traveling)
Average Acceleration
Average acceleration describes how fast the velocity is changing with respect to time. The equation is:
sign determines direction – not the direction the object is traveling, but the direction of the force which is causing the acceleration!
ave
xv t
at t
Sample problem: A motorist drives north at 20 m/s for 20 km and then continues north at 30 m/s for another 20 km. What is his average velocity?
Sample problem: It takes the motorist one minute to change his speed from 20 m/s to 30 m/s. What is his average acceleration?
Average Velocity from a Graph
t
xA
B
x
t
ave
xv
t
Average Velocity from a Graph
ave
xv
t
t
x ABx
t
Average Acceleration from a Graph
t
vA
B
v
t
ave
va
t
• Sample problem: From the graph, determine the average velocity for the particle as it moves from point A to point B.
0
-1
-2
1
2
0 0.1 0.2 0.3 0.4 0.5-3
3
t(s)
x(m)
A
B
• Sample problem: From the graph, determine the average speed for the particle as it moves from point A to point B.
0
-1
-2
1
2
0 0.1 0.2 0.3 0.4 0.5-3
3
t(s)
x(m)
A
B
Lesson
Instantaneous Speed, Velocity, and Acceleration
Warm-up: Describe the motion of the following 4 objects:
t
x
t
x
t
v
t
v
Average Velocity from a Graph
t
x
Remember that the average velocity between the time at A and the time at B is the slope of the connecting line.
AB
Average Velocity from a Graph
t
x
What happens if A and B become closer to each other?
AB
Average Velocity from a Graph
t
x
What happens if A and B become closer to each other?
A B
Average Velocity from a Graph
t
x
AB
What happens if A and B become closer to each other?
Average Velocity from a Graph
t
x
A
B
What happens if A and B become closer to each other?
Average Velocity from a Graph
t
x
A
B
The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is represented by the slope of this tangent line.
A and B are effectively the same point. The time difference is effectively zero.
• Sample problem: From the graph, determine the instantaneous speed and instantaneous velocity for the particle at point B.
0
-1
-2
1
2
0 0.1 0.2 0.3 0.4 0.5-3
3
t(s)
x(m)
A
B
Average and Instantaneous Acceleration
t
v
Average acceleration is represented by the slope of a line connecting two points on a v/t graph.
Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph.
A
B
C
t
x
Instantaneous acceleration is negative where curve is concave down
Instantaneous acceleration is positive where curve is concave up
Instantaneous acceleration is zero where slope is constant
Average and Instantaneous Acceleration
Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What
would the v vs t graph look like?
t
v
Sample problem: Consider an object that is dropped from rest and reaches terminal velocity during its fall. What
would the x vs t graph look like?
t
x
Estimate the net change in velocity from 0 s to 4.0 s
a (m/s2)
1.0
t (s)2.0 4.0
-1.0
Estimate the net displacement from 0 s to 4.0 s
v (m/s)
2.0
t (s)2.0 4.0
Lesson
Derivatives
Sample problem. From this position-time graph determine the velocity at each time interval
x
t
Draw the corresponding velocity-time graph
x
t
Suppose we need instantaneous velocity, but don’t have a graph?
Suppose instead, we have a function for the motion of the particle.
Suppose the particle follows motion described by something like x = (-4 + 3t) m x = (1.0 + 2.0t – ½ 3 t2) m x = -12t3
We could graph the function and take tangent lines to determine the velocity at various points, or…
We can use differential calculus.
Instantaneous Velocity
ave
xv
t
0 0
lim liminst avet t
x dxv v
t dt
Mathematically, velocity is referred to as the derivative of position with respect to time.
Instantaneous Acceleration
0 0
lim lim
ave
avet t
va
tv dv
a at dt
Mathematically, acceleration is referred to as the derivative of velocity with respect to time
Instantaneous Acceleration
Acceleration can also be referred to as the second derivative of position with respect to time.
2
20limt
xd xt
at dt
Just don’t let the new notation scare you; think of the d as a baby , indicating a very tiny change!
Evaluating Polynomial Derivatives
It’s actually pretty easy to take a derivative of a polynomial function. Let’s consider a general function for position, dependent on time.
1
n
n
x At
dxv nAt
dt
Sample problem: A particle travels from A to B following the function x(t) = 2.0 – 4t + 3t2 – 0.2t3.
a) What are the functions for velocity and acceleration as a function of time?
b) What is the instantaneous acceleration at 6 seconds?
Sample problem: A particle follows the function2
4.21.5 5x t
t
a) Find the velocity and acceleration functions.
Hint: look up “reciprocal rule” to find the derivative of the 2nd term!
a) Find the instantaneous velocity and acceleration at 2.0 seconds.
Lesson
Kinematic Graphs -- Laboratory
Lesson
Kinematic Equation Review
Here are our old friends, the kinematic equations
212
2 20 2 ( )
o
o o
v v at
x x v t at
v v a x
Sample problem (basic): Show how to derive the 1st kinematic equation from the 2nd.
•2nd kinematic equation:•x = x0 + v0t + ½ a t2
•Differentiate with respect to time:
•dx/dt = v = v0 + at (1st kinematic equation!)
Sample problem (advanced): Given a constant acceleration of a, derive the first two kinematic equations.
1st equation: derived from definition of average acceleration2nd equation: displacement is equal to the area under the curve of a straight line v-t graph (graph and use geometry)
Draw representative graphs for a particle which is stationary.
x
t
Positionvs
time
v
t
Velocityvs
time
a
t
Accelerationvs
time
Draw representative graphs for a particle which has constant non-zero velocity.
x
t
Positionvs
time
v
t
Velocityvs
time
a
t
Accelerationvs
time
x
t
Positionvs
time
v
t
Velocityvs
time
a
t
Accelerationvs
time
Draw representative graphs for a particle which has constant non-zero acceleration.
Sample problem: A body moving with uniform acceleration has a velocity of 12.0 cm/s in the positive x direction when its x coordinate is
3.0 cm. If the x coordinate 2.00 s later is -5.00 cm, what is the magnitude of the acceleration?
Sample problem: A jet plane lands with a speed of 100 m/s and can accelerate at a maximum rate of -5.00 m/s2 as it comes to a halt.
a) What is the minimum time it needs after it touches down before it comes to a rest?
b) Can this plane land at a small tropical island airport where the runway is 0.800 km long?
Kinematics Graphs Demo
Graph-matching with motion detector
Lesson
Freefall
Free Fall
Free fall is a term we use to indicate that an object is falling under the influence of gravity, with gravity being the only force on the object.
Gravity accelerates the object toward the earth the entire time it rises, and the entire time it falls.
The acceleration due to gravity near the surface of the earth has a magnitude of 9.8 m/s2. The direction of this acceleration is DOWN.
Air resistance is ignored.
Sample problem: A student tosses her keys vertically to a friend in a window 4.0 m above. The keys are caught 1.50 seconds later.
a) With what initial velocity were the keys tossed?
b) What was the velocity of the keys just before they were caught?
Sample problem: A ball is thrown directly downward with an initial speed of 8.00 m/s from a height of 30.0 m. How many seconds later
does the ball strike the ground?
Free Fall Lab