kinematics motion equations 1 constant acceleration constant acceleration problem solving equations...
Post on 22-Dec-2015
266 views
TRANSCRIPT
KinematicsMotion Equations
1
Constant Acceleration
Constant Acceleration Problem Solving
Equations of Motion
Centripetal and Tangential Acceleration
Free-Fall Motion
KinematicsMotion Equations
2
Motion can be determined by using a few simple equations.
rv t
t
0fr r v t
The relationships between velocity and position are (if velocity is constant)
va t
t
0fv v a t
The relationships between acceleration and velocity are (if acceleration is constant)
KinematicsMotion Equations
3
For constant acceleration, we always have three valid equations.
v a t 2
0
1
2r v t a t
Combining these, we find another equation.
tavv f
0
20 0 02f fv v v v v a t a a t
20 0 0 0 0
12 2
2f fv v v v a v t a t v v a r
0 0 2f fv v v v a r
KinematicsMotion Equations
4
We can use these three equations to solve for any motion involving constant acceleration.
v a t
20
1
2r v t a t
0 0 2f fv v v v a r
This equation relates velocity and time.
This equation relates position and time.
This equation relates position and velocity.
KinematicsMotion Equations
5
If we are only dealing with one vector component, then the equations become simple.
v a t
20
1
2r v t a t
0 0 2f fv v v v a r
Let’s just look at the x-component
x xv a t
20
1
2x xx v t a t
2 20 2xf x xv v a x
The y-component and z-component equations are similar.
KinematicsMotion Equations
6
Now let’s see how we use them.
Example: Two race cars are moving on a racetrack. The lead car is ahead by 10 m. Both cars are currently moving at 100 km/hr. If the second car accelerates at 10 m/s2, how long will it take to reach the lead car?
1r
2r
01 10mr
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
2
2 2
m 10
s
fv
a click the icon toopen the worksheet
Microsoft Excel Worksheet
1
1
100 km/hr
0
fv
a
2
KinematicsMotion Equations
7
Now let’s see how we use them.
1r
2r
01 10mr
1 1 1x xv a t 2
1 1 0 1 1 1
1
2x xx v t a t 2 2
1 1 0 1 12xf x xv v a x
2 2 2x xv a t 2
2 2 0 2 2 2
1
2x xx v t a t 2 2
2 2 0 2 22xf x xv v a x
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
2
KinematicsMotion Equations
8
Now let’s see how we use them.
2 2 2x xv a t 2
2 2 0 2 2 2
1
2x xx v t a t
1 1 1x xv a t 2
1 1 0 1 1 1
1
2x xx v t a t 2 2
1 1 0 1 12xf x xv v a x
2 22
km m100 10
hr sxfv t
22 2 22
km 1 m100 10
hr 2 sx t t
22
2 22
km m100 2 10
hr sxfv x
1
km100 0
hrxfv
1 1
km100
hrx t
22
1
km100
hrxfv
2
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
1r
2r
01 10mr
2 22 2 0 2 22xf x xv v a x
KinematicsMotion Equations
9
Now let’s see how we use them.
2 22
km m100 10
hr sxfv t
22 2 22
km 1 m100 10
hr 2 sx t t
22
2 22
km m100 2 10
hr sxfv x
1
km100 0
hrxfv
1 1
km100
hrx t
22
1
km100
hrxfv
ttt 21
2 1 01x x x
2 2
km m100 10
hr sxfv t
22 2
km 1 m100 10
hr 2 sx t t
22
2 22
km m100 2 10
hr sxfv x
1
km100 0
hrxfv
2
km10 m 100
hrx t
22
1
km100
hrxfv
2
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
1r
2r
01 10mr
KinematicsMotion Equations
10
Now let’s see how we use them.
ttt 21
1 2
km m100 10
hr sxfv t
21 2
km 1 m100 10
hr 2 sx t t
22
1 12
km m100 2 10
hr sxfv x
2
km100 0
hrxfv
1
km10 m 100
hrx t
22
2
km100
hrxfv
2
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
1r
2r
01 10mr
2 1 01x x x
KinematicsMotion Equations
11
Now let’s see how we use them.
ttt 21
21 2
km 1 m100 10
hr 2 sx t t
1
km10 m 100
hrx t
22
km km 1 m10 m 100 100 10
hr hr 2 st t t
2
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
1r
2r
01 10mr
2 1 01x x x
KinematicsMotion Equations
12
Now let’s see how we use them.
ttt 21
22
km km 1 m10 m 100 100 10
hr hr 2 st t t
22
1 m10 m 10
2 st
2
2
10 m2
m10
s
t
1.44 st
2
10
1
100 km/hr
0
v
a
20
2 2
100 km/hr
m 10
s
v
a
1r
2r
01 10mr
2 1 01x x x
KinematicsMotion Equations
13
What Happened to Centrifugal Force?
There is no such thing as centrifugal force.
So where did it come from?
A mistaken assumption is made that the forces on particles moving in a circle with constant speed have no forces acting on them.
Why is this false?
Acceleration comes from changes in velocity (direction, not just speed).
Circular motion requires acceleration and thus requires force.
KinematicsMotion Equations
14
centifugalF
T
What Happened to Centrifugal Force?
The myth starts from the mistaken idea that there is no acceleration and therefore the total force is zero.
0centifugal amFT
KinematicsMotion Equations
15
What Happened to Centrifugal Force?
The fact is that velocity is changing and the acceleration is the centripetal acceleration. The force is NOT zero!!!!
lcentripetaamT
T
KinematicsMotion Equations
16
Centripetal vs. Tangential Acceleration
Centripetal acceleration causes a particle to change its direction.
2
c
va
r
It points toward the center of the circle
r
v
KinematicsMotion Equations
17
Centripetal vs. Tangential Acceleration
Tangential acceleration causes a particle to change its speed.
t
va
t
It points along the tangent to the line of motion.
KinematicsMotion Equations
18
Free Fall
Any particle, subject only to the force of gravity is in free-fall.
If an object is in free-fall and we define the positive y-axis as upward, then its acceleration is always given by
where g is the acceleration due to gravity and has a value of 9.81 m/s2 near the surface of the earth.
Note that the acceleration parallel to the earth’s surface is zero.
j ga
KinematicsMotion Equations
19
Particle’s in free fall are subject only to the force of gravity.
Every particle in free-fall has an acceleration of 9.81 m/s2 downward.
Free Fall
The motion diagram for any object in free-fall that starts from rest is the same.
KinematicsMotion Equations
20
For particle’s that do not start at rest…
1.The vertical acceleration is 9.81 m/s2 downward.2.The horizontal acceleration is zero. (The horizontal velocity is constant.)
In other words, objects move in a very predictable way.But then, you already know this.
Free Fall
KinematicsMotion Equations
21
The motion of a baseball hit at an angle, undergoing free fall is a parabola.
Free Fall
KinematicsMotion Equations
22
The motion of a rocks thrown from a cliff at different horizontal speeds has some similarities.
Free Fall
KinematicsMotion Equations
23
Equations
In free fall, the vertical position, velocity and acceleration are related by the equations
the horizontal position, velocity and acceleration are related by the equation
Note that we can write the components of the initial velocity as
20 2
1gttvy y gtvy
tvx x0 0 xv
000 cosvv x 000 sinvv y