Mechanical Rate (a.k.a. Motion)

TEKS TLW knows the laws governing motion (TEKS 7) TEKS 7.A - Generate and interpret relevant equations using graphs and charts for one- and two-dimensional motion i.One dimensional equations for: displacement, distance, speed, velocity, average velocity, acceleration, and average acceleration ii.Two-dimensional equations for projectile and circular motion iii.Using and describing vector forces and resolution

TEKS TLW knows the laws governing motion (TEKS 7) TEKS 7.B – Describe and calculate the effects of forces on motion of objects including law of inertia, impulse, and conservation of momentum TEKS 7.C – Develop and interpret a free-body diagram TEKS 7.D – Identify and describe motion relative to different frames of reference

Lesson Plan Objectives Identify and describe motion relative to different frames of reference – such as heights and orbits Use real-time technology (photo-gates, ramps, stop watchers, etc.) in hands-on labs Prepare and interpret graphs from data collected on linear, projectile, and circular motion Define Speed, velocity, and acceleration. Explain the difference between speed and velocity. Explain the difference between velocity and acceleration. Use speed, velocity, and acceleration to solve problems involving linear (one-dimensional) motion.

Lesson Plan Objectives Use speed, velocity, and acceleration to solve problems involving projectile motion Define angular speed and angular acceleration. Use angular speed and angular acceleration to solve problems involving rotational motion. Create and use free-body diagrams to analyze force scenarios

Speed Speed is the ratio between the distance traveled and the elapsed time. (scalar quantity) Distance traveled (  d) Time interval (  t) Speed = Examples of units to use: m/sec, km/hr, ft/sec, or miles/hr

Average Speed When speed varies between point A and B Distance traveled (  d) Time interval (  t) Average Speed = dd tt s avg = = d 2 – d 1 t 2 –t 1 d 2 = final distance (d f ) d 1 = initial starting point (d i ) t 2 = final time (t f ) t 1 = initial time (t i )

Velocity A vector quantity giving the speed (magnitude) and direction of travel. distance time Velocity = dd tt v avg =

Acceleration Describes the rate of change of an object’s velocity Velocity change (  v) Time interval (  t) Average Acceleration = vv tt a avg == v 2 – v 1 t 2 –t 1 Negative acceleration is called deceleration v 2 = final velocity (v f ) v 1 = initial velocity (v i ) t 2 = final time (t f ) t 1 = initial time (t i )

Free Fall Definition – the movement of an object in response to gravitation attraction As an object falls towards Earth it will accelerate at a constant rate of 9.8 m/s 2 – regardless of mass ( g = 9.8 m/s 2 ) It is common to neglect air resistance in high school curriculum… but it does play a part in real life Downward acceleration will be positive, upward will be represented by a negative. …. Likewise, upward velocity will be negative and upward direction will be negative.

Free Fall Displacement of falling object = Δd = v initial Δt + (1/2)g Δt 2 Final velocity of falling object = v final 2 = v initial 2 + 2g Δd OR Final velocity of falling object = v final = v initial + g Δt

Free Fall Time for object to reach Earth = Δt = v final - v initial g OR Time for object to reach Earth = Δt = 2 Δd g

Projectile Motion A projectile is any object upon which the only force is gravity Projectiles travel with a parabolic trajectory due to the influence of gravity There are no horizontal forces acting upon projectiles, and thus no horizontal acceleration

Projectile Motion Projectiles always maintain a constant horizontal velocity (neglecting air resistance) Projectiles always experience a constant vertical acceleration of 9.8 m/s 2 downward (neglecting air resistance). Horizontal and vertical motion are completely independent of each other. (i. e. – there are horizontal and vertical components to velocity)

Projectile Motion For an object beginning and ending at the same height it takes the same amount of time to reach highest point as it does to return to original position Objects dropped from a moving vehicle have the same velocity as that vehicle

Projectile Motion horizontal distance (d x ) in meters = v x t vertical distance (d y ) in meters = v y ∆t + 1/2g∆t 2 vertical distance (d y ) at an angle in meters = v y (sin θ) ∆t + 1/2g∆t 2

Projectile Motion angular range (R) in meters = v i 2 sin 2 θ g hang time (t) in seconds = 2v y (sin θ) g

Angular Speed Rate of rotational motion. angular displacement (  ) time interval (  t) (  ) = Angular Speed omega 1 revolution = 360º = 2  radians radians or revolutions

Angular Acceleration Ration of the change in angular speed to the time interval. Angular speed change (  ) Time interval (  t) Angular Acceleration =  tt a ==  2 –  1 t 2 –t 1

Summary Speed is a measure of the rate of motion of an object. It is the ratio of distance traveled to the time interval. Speed is a scalar quantity. Velocity is the ratio of displacement to the time interval. Velocity and displacement are vector quantities. Speed is the magnitude of velocity. Acceleration is a measure of the rate of change of an object’s velocity. It is the ratio of change in velocity to the time interval. Angular speed is a measure of the rate of rotational motion of an object. It is the ratio of angular displacement to time interval. Angular acceleration is a measure of the rate of change of an object’s angular speed. It is the ratio of change in angular speed to the time interval.