1. Linear Kinematics Describing Objects in Linear Motion
Chapter 2
Linear Kinematics
Describing Objects in Linear Motion

3. Objectives Distinguish between linear, angular, and general motion
Define distance traveled and displacement, and distinguish between the two
Define average speed and average velocity, and distinguish between the two
Define instantaneous speed and instantaneous velocity

4. Objectives Define average acceleration
Define instantaneous acceleration
Name the units of measurement for distance traveled and displacement, speed and velocity, and acceleration
Use the equations of projectile motion to determine the vertical or horizontal position of a projectile given the initial velocities and time

5. What is Motion? Action or process of a change in position
Two things are necessary for motion to occur: space to move in and time during which to move
Movements can be classified as linear, angular, or both (general)

6. Linear Motion Referred to as translation
Occurs when all points on a body or object move the same distance, in the same direction, and at the same time
Rectilinear and Curvilinear translation—Difference is that the paths followed by the points on an object in curvilinear translation are curved, so the direction of motion of the object is constantly changing, even though the orientation of the object does not change

7. Angular Motion Referred to as rotary motion or rotation
Occurs when all points on a body or object move in circles (or parts of circles) about the same fixed central line or axis
Can occur about an axis within the body (ice skater in a spin) or outside the body (child on a swing)

8. General Motion Combination of linear and angular motion
Most common type of motion exhibited in sports and human movement
Running and Bicycling—Linear motion as a result of the angular motion of the legs and arms
General motion of a body or object can be broken down into linear and angular components

9. Linear Kinematics Concerned with the description of linear motion
Position—Location in space
Where is an object in space at the beginning or end of its movement or at some time during its movement
Consider the importance of the position of players for the strategies employed in sports such as football, tennis, racquetball, squash, soccer, ect…

10. Position
Can be described in one (x), two (x, y), or three (x, y, z) dimensions
To describe the position of an object, a Cartesian coordinate system is used
Need to identify a fixed reference point to serve as the origin of the coordinate system
Number of axes corresponds to the number of dimensions
Axes must be at right angles when describing the position of an object in two or three dimensions
Each axis has a positive and negative direction

14. Distance Traveled and Displacement
Distance—Length of the path followed by the object whose motion is being described, from the starting position to the ending position
Displacement—Straight-line distance in a specific direction from the starting position to the ending position
Resultant displacement—Distance measured in a straight line from the starting position to the ending position

16. Distance Traveled and Displacement
Football example:
Resultant displacement can be resolved into components in the x-direction (across the field) and y-direction (down the field toward the goal)
In this case the y-displacement of the running back is the measure of importance
We can find the y-displacement by subtracting his initial position from his final position

17. Distance Traveled and Displacement
dy = Δy = yf – yi
dy = displacement in the y-direction
Δy = change in y-position
yf = final y-position
yi = initial y-position
dy = Δy = yf – yi = 35yd – 5yd
dy = +30yd

18. Distance Traveled and Displacement
We may also be curious about the player’s displacement across the field (in the x-direction)
dx = Δx = xf – xi
dx = displacement in the x-direction
Δx = change in x-position
xf = final x-position
xi = initial x-position
dx = Δx = xf – xi = 5yd – 15yd
dx = -10yd

19. Distance Traveled and Displacement
Resultant displacement could be found using trigonometric relationships
Hypotenuse of right triangle represents resultant displacement
A2 + B2 = C2 or (Δx)2 + (Δy)2 = R2
(-10)2 + (30)2 = R2
100yd yd2 = R2
1000yd2 = R2
R = √1000yd2
R = 31.6yd

20. Distance Traveled and Displacement
To find the direction of the resultant displacement we can use the relationship between the two sides of the displacement triangle
Tanθ = opposite side/adjacent side
θ = arctan (opposite side/adjacent side)
θ = arctan (Δx/Δy)
θ represents the angle between the resultant displacement vector and the y-displacement vector
θ = arctan (-10yd/30yd)
θ = arctan (-.3333) = 18.4°

21. Speed and Velocity Speed—Rate of motion
Velocity—Rate of motion in a specific direction
Average speed—Distance traveled divided by the time it took to travel that distance
s = ℓ/Δt
s = average speed
ℓ = distance traveled
Δt = time taken or change in time

22. Speed and Velocity
SI unit for describing speed is meters per second (other units commonly used include miles per hour or kilometers per hour)
Average speed does not tell us what went on during the race itself
Does not tell us the maximum speed reached by the racer
Does not indicate when the racer was speeding up or slowing down

24. Speed and Velocity Johnson: Average speed entire 100m s = 100m/9.79s
Average speed first 50m
s = 50m/5.50s
s = 9.09m/s
Average speed 50m to 100m
s = 100m-50m/9.79s – 5.50
s = 50m/4.29s
s = 11.66m/s
Lewis:
Average speed entire 100m
s = 100m/9.92s
s = 10.08m/s
Average speed first 50m
s = 50m/5.65s
s = 8.85m/s
Average speed 50m to 100m
s = 100m-50m/
s = 50m/4.27s
s = 11.71m/s

25. Speed and Velocity
10m split times can be used to determine the average speed of each sprinter during each 10m interval (e.g. during the 50-60m interval both sprinters reached maximum speed)
By taking more split times during the race, we can determine the runners’ average speeds for more intervals and shorter intervals
Instantaneous speed or velocity—Speed or velocity of an object at a specific instant in time (very small time interval)
Baseball radar gun

27. Velocity
Average velocity—Displacement of an object divided by the time it took for that displacement
v = d/Δt
v = average velocity
d = displacement
Δt = time taken or change in time

28. Velocity
If the motion of the object under analysis is in a straight line and rectilinear, with no change in direction, average speed and average velocity will be identical in magnitude
However, if the direction of motion changes, speed and the magnitude of velocity are not synonymous (e.g. 100m swim in a 50m pool)

29. Importance of Speed and Velocity
Direct indicators of performance (e.g. baseball, soccer, track and field)
Baseball example:
In his prime Nolan Ryan could pitch the ball 101mi/hr or 148ft/s or 45m/s
Distance from the pitching rubber to home plate is 60.5ft (ball is released ~2.5ft in front of the rubber), so the horizontal distance it must travel to reach the plate is 58ft
How much time does a batter have to react to Nolan Ryan’s fastball?

30. Importance of Speed and Velocity
Sample Problem 2.1 (text p. 61)
Average horizontal velocity of a penalty kick in soccer is 22m/s
Horizontal displacement of the ball from the kicker’s foot to the goal is 11m
How long does it take for the ball to reach the goal after it is kicked?

31. Acceleration Rate of change in velocity
An object accelerates if the magnitude or direction of its velocity changes (speeds up, slows down, or changes direction)
Average acceleration—Change in velocity divided by the time it took for that velocity change to take place
Instantaneous acceleration—Rate of change in velocity at a specific instant in time
SI unit for describing acceleration are meters per second per second or m/s2

32. Acceleration a = Δv/Δt a = vf – vi/Δt a = average acceleration
Δv = change in velocity
vf = instantaneous velocity at the end of an interval of final velocity
vi = instantaneous velocity at the beginning of an interval, or initial velocity
Δt = time taken or change in time

33. Acceleration
Direction of motion is not necessarily the same as the direction of the acceleration
Before analyzing a problem, first establish which direction “+” will be assigned to

34. Acceleration
If final velocity is less than initial velocity, the change in velocity is a negative number and the resulting average acceleration is negative (slowing down in the positive direction)
Negative acceleration will also result if the initial and final velocities are both negative and if the final velocity is a larger negative number than the initial velocity (object is speeding up in the negative direction)

36. Acceleration Car example: Car can accelerate from 0 to 60 in 7s
a = Δv/Δt
a = vf – vi/Δt
a = 60mi/hr – 0mi/hr/7s
a = 8.6mi/hr/s
In 1s, this car’s velocity increases (speeds up) by 8.6m/hr
If the car is accelerating at 8.6mi/hr/s and moving at 30mi/hr, how fast will the car be traveling 1s, or 2s later?

37. Uniform Acceleration and Projectile Motion
In certain situations the acceleration of an object is constant—Occurs when the net external force acting on an object is constant (e.g. gravity)
If an object undergoes uniform acceleration, its position and velocity at any instant in time can be predicted

38. Vertical Motion of a Projectile
Projectile—Object that has been projected into the air or dropped and is only acted on by the forces of gravity and air resistance
Acceleration due to gravity or g is 9.81m/s2 downward (upward is the positive direction)
We can predict projectile velocity and position

39. Vertical Motion of a Projectile
Equations to describe the vertical motion of a projectile:
Vertical position of a projectile:
yf = yi + viΔt + ½g(Δt)2
Vertical velocity of projectile:
vf = vi + gΔt
vf2 = vi2 + 2gΔy

40. Vertical Motion of a Projectile
yi = initial vertical position
yf = final vertical position
Δy = yf – yi = vertical displacement
Δt = change in time
vi = initial vertical velocity
vf = final vertical velocity
g = acceleration due to gravity = -9.81m/s2

41. Vertical Motion of a Projectile
If we are analyzing the motion of something that is dropped, the equations are simplified
vi = 0
yi = 0
Vertical position of falling object:
yf = ½g(Δt)2
Vertical velocity of falling object:
vf2 = 2gΔy

43. Vertical Motion of a Projectile
Sample Problem 2.2 (text p. 66)
Upward velocity of the ball as it passes any height on the way up is the same as the downward velocity of the ball when it passes that same height on the way down
Time it takes for the ball’s upward velocity to slow down to zero is the same as the time it takes for the ball’s downward velocity to speed up from zero to the same size velocity downward
Δtup = Δtdown (if the initial and final y-positions same)
Δtflight = 2Δtup (if the initial and final y-positions same)
Sample Problem 2.3 (text p. 67)

44. Horizontal Motion of a Projectile
Difficult to examine or observe the horizontal motion of a projectile separately from its vertical motion, though, because you see both horizontal and vertical motions simultaneously
Horizontal velocity of a projectile is constant, and its horizontal motion is in a straight line (ignoring air resistance)

46. Horizontal Motion of a Projectile
Equations to describe the horizontal motion of a projectile:
Horizontal position of projectile:
xf = xi + vΔt
x = vΔt—if the initial position is zero
Horizontal velocity of projectile:
v = vf = vi = constant (ignoring air resistance)

47. Horizontal Motion of a Projectile
xi = initial horizontal position
xf = final horizontal position
Δt = change in time
vi = initial horizontal velocity
vf = final horizontal velocity

48. Linear Kinematics
Vertical and horizontal motions of a projectile are independent of each other
An equation can be derived to describe the path of a projectile in horizontal and vertical dimensions
yf = yi + vyi (x/vx) + ½g(x/vx)2
Equation of a parabola—Describes the vertical (y) and horizontal (x) coordinates of a projectile during its flight based solely on the initial vertical position and vertical and horizontal velocities

50. Projectiles in Sport
Any ball used in sports becomes a projectile when it is thrown, released, hit, kicked, ect…
Once in flight the path of the ball cannot be changed (ignoring air resistance)
Vertically constant acceleration downward
Horizontally ball won’t slow down or speed up
Initial conditions determine projectile motion

51. Projectiles in Sport
In sports involving projectiles, one of three things concerned with:
Time of flight—Dependent on initial vertical velocity and initial vertical position
Football punt, tennis lob, gymnastics, diving
Optimum angle of projection to achieve maximum height and time of flight is 90° (straight up)
Sometimes desirable to minimize flight time—Volleyball spike, tennis overhead smash, baseball throw, soccer penalty kick—Projection angle < 45°

52. Projectiles in Sport
Peak height reached—Dependent on initial vertical velocity and initial vertical position
Higher the projectile is at release and the faster it is moving upward at release, the higher it will go (basketball and volleyball players)
High jumping—Angle of projection > 45°

53. Projectiles in Sport
Horizontal displacement—Dependent on initial horizontal velocity, initial vertical velocity, and initial height
Shot put, hammer throw, discuss throw, javelin throw, and long jump
Javelin and discuss can be affected by air resistance so equations may not be accurate
Maximum horizontal displacement will occur if the horizontal and vertical components of the initial velocity are equal, or when the projection angle is 45° (assuming initial release height is zero)
The higher the release height and the greater the lift effects of air resistance on the projectile, the farther below 45° the projection angle should be (shot put 35°)

55. Summary
Motion can be classified as a combination of linear, angular, or a combination of both (general motion)
Displacement, velocity, and acceleration are vector quantities described by size and direction
Horizontal velocity of a projectile is constant and its vertical velocity changes at the rate of 9.81m/s2