Chapter 2 Linear Kinematics Describing Objects in Linear

Chapter 2 Linear Kinematics Describing Objects in Linear Motion

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

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

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)

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

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)

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

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…

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

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

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

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 = 35 yd – dy = +30 yd

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 = 5 yd – 15 yd dx = -10 yd

Distance Traveled and Displacement • Resultant displacement could be found using trigonometric relationships – Hypotenuse of right triangle represents resultant displacement – A 2 + B 2 = C 2 or (Δx)2 + (Δy)2 = R 2 – (-10)2 + (30)2 = R 2 – 100 yd 2 + 900 yd 2 = R 2 – 1000 yd 2 = R 2 – R = √ 1000 yd 2 – R = 31. 6 yd

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 (-10 yd/30 yd) – θ = arctan (-. 3333) = 18. 4°

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

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

Speed and Velocity • • Johnson: Average speed entire 100 m s = 100 m/9. 79 s s = 10. 21 m/s • • Lewis: Average speed entire 100 m s = 100 m/9. 92 s s = 10. 08 m/s • Average speed first 50 m • s = 50 m/5. 50 s • s = 9. 09 m/s • Average speed first 50 m • s = 50 m/5. 65 s • s = 8. 85 m/s • • Average speed 50 m to 100 m s = 100 m-50 m/9. 79 s – 5. 50 s = 50 m/4. 29 s s = 11. 66 m/s Average speed 50 m to 100 m s = 100 m-50 m/9. 92 -5. 65 s = 50 m/4. 27 s s = 11. 71 m/s

Speed and Velocity • 10 m split times can be used to determine the average speed of each sprinter during each 10 m interval (e. g. during the 50 -60 m 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

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

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. 100 m swim in a 50 m pool)

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 101 mi/hr or 148 ft/s or 45 m/s – Distance from the pitching rubber to home plate is 60. 5 ft (ball is released ~2. 5 ft in front of the rubber), so the horizontal distance it must travel to reach the plate is 58 ft – How much time does a batter have to react to Nolan Ryan’s fastball?

Importance of Speed and Velocity • Sample Problem 2. 1 (text p. 61) – Average horizontal velocity of a penalty kick in soccer is 22 m/s – Horizontal displacement of the ball from the kicker’s foot to the goal is 11 m – How long does it take for the ball to reach the goal after it is kicked?

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 or m/s 2

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

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

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)

Acceleration • Car example: – Car can accelerate from 0 to 60 in 7 s – – – a = Δv/Δt a = vf – vi/Δt a = 60 mi/hr – 0 mi/hr/7 s a = 8. 6 mi/hr/s In 1 s, this car’s velocity increases (speeds up) by 8. 6 m/hr – If the car is accelerating at 8. 6 mi/hr/s and moving at 30 mi/hr, how fast will the car be traveling 1 s, or 2 s later?

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

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. 81 m/s 2 downward (upward is the positive direction) • We can predict projectile velocity and position

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 • vf 2 = vi 2 + 2 gΔy

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. 81 m/s 2

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: – vf 2 = 2 gΔy

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)

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)

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)

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

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

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

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°

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°

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°)

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. 81 m/s 2