Understanding Motion

1. Understanding Motion Linear Motion

2. Motion • The motion of an object can only be recognized when something is established as a basis of comparison…a reference point • We say an object is moving when its position changes compared to that reference point • For most day-to-day situations, the Earth, and those things affixed to it, serve as a convenient reference frame.

3. Position and Time • These are the two most fundamental physical quantities that can be measured to describe an object’s motion. • The relationship between these variables can be discovered experimentally and modeled using mathematics in both graphical and equation form.

4. Position vs. Time Graphs • The magnitude (size) of the slope tells us… • The algebraic sign of the slope tells us… • The magnitude and sign together tell us… • The vertical intercept tells us… • When this graph is a straight line we know…

5. Position vs. Time Graphs The generic equation for a linear graph is… y = mx + b In terms of the physical quantities being plotted this becomes… x = mt + b Position, x (m) Time, t (s)

6. Position vs. Time Graphs If we replace the slope and intercept terms with what they tell us we get… x = vt + xo Where v is the velocity (m/s) and xo is the initial position (m) of the object in motion m = v Position, x (m) xi Time, t (s)

7. x = vt + xo • This equation (straight line with slope v and intercept xo) is a model that describes the relationship between position and time for an object moving with constant velocity. x = position of object after time, t v = velocity of object (speed in a direction) t = elapsed time xo = starting position of the object

8. Ball on a ramp questions: • What information does the slope of an x-t graph tell us? • Does your x-t graph for the ball on a ramp have slope? • What does the shape of your x-t graph tell us about the motion of the ball? • Does the information from your v-t graph support the description given in #3? How so? • What information does the slope of a v-t graph tell us? • What information does the intercept of the v-t graph tell us? • What value would you expect for the intercept in this lab activity? Why?

9. Constantly Accelerated Motion(ball on a ramp) • Velocity-time graphs are linear… v  t • Slope is constant  The rate at which velocity is changing is constant SLOPE = ACCELERATION • Position-time graphs are NOT linear, they are quadratic… x  t2 • slope is NOT constant  Velocity is changing

10. vf= at + vo • This equation is a model describing the relationship between velocity and time for an object that is constantly accelerating vf– “final” velocity after time, t a – acceleration (slope of v-t graph) t – elapsed time vo – starting velocity (intercept of v-t graph)

11. x = ½ at2 + vot + xo • This equation models the relationship between position and time for constantly accelerated motion • This equation emerges from our ball on a ramp data or it could be derived (we will!) x = position after time, t a = acceleration vo = starting velocity t = elapsed time xo = starting position

12. A summary of motion equations thus far… Constant velocity: x = vt+ xox = vt or v= x/t (v is constant or average velocity) Constant acceleration: vf= at + vo a = v/t x = ½ at2 + vot + xo x = ½ at2 + vot