Kinematics : Kinematics Kinematics is the study of motion
Motion involves three concepts
Displacement
Velocity
Acceleration
These concepts can be used to study objects in motion
Kinematics is not concerned with the cause of the motion
Position and Displacement : Position and Displacement Fixed reference point
X-axis used for horizontal motion
Y-axis used for vertical motion
Change from one position to another is displacement
Displacement is vector quantity
Slide 3 : A skater glides along a circular path of radius 5.00 m. If he glides around half of the circle, determine;
the magnitude of the displacement vector
the distance traveled, and
the magnitude of the displacement if he skates all the way around the circle.
Slide 4 : Start Finish Displacement = 10 m
Distance travelled = pie x radius = 15.7 m
Displacement = 0 m
Average Speed vs Average Velocity : Average Speed vs Average Velocity
Instantaneous Velocity : Instantaneous Velocity Slope of the tangent to the displacement-time graph at a particular time.
Acceleration : Acceleration Rate of change of velocity of an object.
Scalar quantity
Acceleration & velocity in same direction, speed increases
Acceleration and velocity in opposite direction, speed decreases
1D motion with constant acceleration : 1D motion with constant acceleration We will consider the case of constant acceleration only
Use equations of motion
Slide 9 : Example A speedboat increases its speed uniformly from 20 m/s to 30 m/s in a distance of 200 m. Find
the magnitude of its acceleration and
the time it takes the boat to travel the 200-m distance.
Solution
The unknown acceleration is constant. The boat is initially moving at vi = 20 m/s has a displacement s = 200 m and accelerates to vf = 30 m/s.
Slide 10 : Solution (cont’d) We find the acceleration from
The required time may be found from
Slide 11 : Another Example A Cessna aircraft has a lift-off speed of 120 km/h.
What minimum constant acceleration does the aircraft require if it is to be airborne after a takeoff runway of 240 m?
How long does it take the aircraft to become airborne?
Free Falling Objects : Free Falling Objects All objects moving under the influence of gravity only are said to be in free fall
Free fall does not depend on the object’s original motion
All objects falling near the earth’s surface fall with a constant acceleration
The acceleration is called the acceleration due to gravity, and indicated by g
Acceleration due to Gravity : Acceleration due to Gravity Symbolized by g
g = 9.80 m/s²
g is always directed downward
toward the center of the earth
Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion
Free Fall – an object dropped : Free Fall – an object dropped Initial velocity is zero
Let up be positive
Use the kinematic equations
Acceleration
g = -9.80 m/s2 vi= 0
a = g
Example : Example A ball at rest released from the top of a building, hits the ground after falling for 6 s. What is the height of the building? Ignore air resistance
Solution : Solution Since vi = 0 and a = g the equation reduces to Negative represents displacement was downward same as the height of building of 176.4 m
Free Fall – an object thrown downward : Free Fall – an object thrown downward a = g = -9.80 m/s2
Initial velocity ? 0
With upward being positive, initial velocity will be negative Vi = 0
Free Fall - object thrown upward : Free Fall - object thrown upward Initial velocity is upward, so positive
The instantaneous velocity at the maximum height is zero
a = g = -9.80 m/s2 everywhere in the motion
The motion may be symmetrical
Then tup = tdown
Then v = -vo
If motion is not symmetrical
Break the motion into various parts
Generally up and down
Non-symmetrical Free Fall : Non-symmetrical Free Fall Need to divide the motion into segments
Possibilities include
Upward and downward portions
The symmetrical portion back to the release point and then the non-symmetrical portion
Study examples 2.8 – 2.9
Two Dimensional Motion with Constant Acceleration – Projectile Motion : Two Dimensional Motion with Constant Acceleration – Projectile Motion
Slide 21 : vx vy Max. height
v = vx
vy = 0
An Example : An Example An Alaskan rescue plane drops a package of emergency rations to a stranded hiker. The plane is travelling horizontally at 40 m/s at a height of 100 m above ground.
Where does the package strike the ground relative to the point at which it was released?
What are the horizontal and vertical components of the velocity of the package just before it hits the ground?
Example 2 : Example 2 A long jumper leaves the ground at an angle of 20o to the horizontal and at a speed of 11 m/s.
How long does it take for him to reach the maximum height?
What is the maximum height?
How far does he jump?
Relative Velocity : Relative Velocity Relative velocity is about relating the measurements of two different observers
It may be useful to use a moving frame of reference instead of a stationary one
It is important to specify the frame of reference, since the motion may be different in different frames of reference
There are no specific equations to learn to solve relative velocity problems
Slide 25 : Relative Velocity Notation The pattern of subscripts can be useful in solving relative velocity problems
Assume the following notation:
E is an observer, stationary with respect to the earth
A and B are two moving cars
Slide 26 : Relative Position Equations is the position of car A as measured by E
is the position of car B as measured by E
is the position of car A as measured by car B
Slide 27 : Relative Position The position of car A relative to car B is given by the vector subtraction equation
Slide 28 : Relative Velocity Equations The rate of change of the displacements gives the relationship for the velocities BE
Example : Example A boat is heading due north as it crosses a wide river with a velocity of 10.0 km/h relative to the water. The river has a uniform velocity of 5 km/h due east. Determine the velocity of the boat with respect to an observer on the river bank.
Another Example : Another Example The pilot of an aircraft wishes to fly due west in a 50.0-km/h wind blowing toward the south. If the speed of the aircraft relative to the air is 200 km/h,
in what direction should the aircraft head, and
what will be its speed relative to the ground?