2-1 Displacement and Velocity Chapter 2 Motion in One Dimension 2-1 Displacement and Velocity Motion – is a change in position measured by distance and time. Object’s change in position is relative to a reference point Kinematics = part of physics that describes motion without discussing the forces that cause the motion Displacement vs distance traveled Displacement = the length of the straight-lined path between two points – not the total distance traveled.

Displacement is not always equal to the distance traveled! Displacement can be positive or negative! Question: Does the odometer in your car measure distance or displacement?

Displacement can be positive or negative values During which time intervals did it travel is a positive direction? What about a negative direction?

Displacement can be positive or negative values During which time intervals did it travel is a positive direction? What about a negative direction? Positive direction: 0-10 sec and 40 and 60 sec Negative direction 15-40 sec

Draw a position versus time graph for each of the following: constant forward motion constant backward motion constant acceleration constant deceleration sitting still Time (s) Position (m)

constant forward motion Straight sloped line going higher [slope (therefore velocity) does not change] Time (s) Position (m)

constant backward motion Straight sloped line going lower [slope (therefore velocity) does not change] Time (s) Position (m)

constant acceleration Increasing slope [slope (therefore velocity) increases] Time (s) Position (m)

constant deceleration Decreasing slope [slope (therefore velocity) decreases] Time (s) Position (m)

sitting still Straight line with no slope Time (s) Position (m)

Position-Time Graphs Slope on a position time graph is velocity. Time (s) Position (m) Rise (ΔY) Run (Δ X) Slope = ___________ Δ Displacement (ΔY) Δ Time (Δ X) Velocity = _______________ (ΔY) (Δ X) Therefore: slope of P-T graph = velocity

Motion in 1 Dimension Use the symbol ‘x’ for horizontal motion and “y” for vertical motion (up and down). Change in position (distance) = ∆X or ∆Y Greek letter delta (∆) = a change in position (∆) = (final ) – (initial) Formula for Displacement:

V = ∆X /∆t (with direction) SI units = m/s Difference between Velocity and Speed Velocity describes motion with both a direction and a numerical value (a magnitude). Example: Moving at 65 mph due North Speed has no direction, only magnitude. 65 mph Speed: S = ∆X /∆t Velocity: V = ∆X /∆t (with direction) SI units = m/s

Average velocity is the total displacement divided by the time interval during which the displacement occurred. Vavg = Vi + Vf 2 And if Vi = 0, then, Vavg = ½ vf

Finding Average Velocity if Velocity is not constant 10 9 D 8 i s 7 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Average Velocity if Velocity is not constant

* * Finding Average Velocity - pick 2 points 10 9 D 8 i s 7 p l 6 a m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Average Velocity - pick 2 points * *

* * Finding Average Velocity - draw a line between them 10 9 D 8 i s 7 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Average Velocity - draw a line between them * *

* * Finding Average Velocity - find their coordinates 10 9 D 8 i p l 6 a c 5 e m 4 n 3 t 2 (m) (0 , 0) 1 0 1 2 3 4 Time (s) Finding Average Velocity - find their coordinates * *

* * Finding Average Velocity ∆t tf - ti 10 9 - calculate the slope D 8 m 4 n 3 t 2 (m) (0 , 0) 1 0 1 2 3 4 Time (s) Finding Average Velocity - calculate the slope ∆x xf - xi ∆t tf - ti * *

* * Finding Average Velocity ∆t tf - ti 3.5 - 0 10 9 s 7 (3.5 , 7) p l 6 a c 5 e m 4 n 3 t 2 (m) (0 , 0) 1 0 1 2 3 4 Time (s) Finding Average Velocity - calculate the slope ∆x xf - xi 7 - 0 ∆t tf - ti 3.5 - 0 * *

* * 2 Finding Average Velocity ∆t tf - ti 3.5 - 0 10 9 i m/s s 7 (3.5 , 7) p l 6 a c 5 e m 4 n 3 t 2 (m) (0 , 0) 1 0 1 2 3 4 Time (s) Finding Average Velocity - calculate the slope ∆x xf - xi 7 - 0 ∆t tf - ti 3.5 - 0 2 * *

Instantaneous Velocity We can also calculate the velocity of a moving object at any point along the curve. This is called the - Instantaneous Velocity Draw a line tangent to the velocity curve, and find its slope – Speedometer

Formula: Instantaneous Velocity Vinst x2 - x1 ∆xtan = = t2 - t1 ∆ttan

The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position-versus-time graph.

10 9 D 8 i s 7 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) .

Finding Instantaneous Velocity 10 9 D 8 i s 7 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - draw the tangent line

* * Finding Instantaneous Velocity 10 9 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - find 2 convenient points – far away from each other * *

* * 10 Finding Instantaneous Velocity 9 - find their coordinates D 8 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 (2 , 1) 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - find their coordinates * *

* * 10 Finding Instantaneous Velocity ∆t t2 - t1 9 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 (2 , 1) 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - calculate the slope ∆x x2 - x1 ∆t t2 - t1 * *

* * 10 Finding Instantaneous Velocity ∆t t2 - t1 4 - 2 9 p l 6 a c 5 e m 4 n 3 t 2 (m) 1 (2 , 1) 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - calculate the slope ∆x x2 - x1 7 - 1 ∆t t2 - t1 4 - 2 * *

* * 10 3 Finding Instantaneous Velocity 9 D 8 i m/s (4 , 7) s 7 p l 6 2 (m) 1 (2 , 1) 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - calculate the slope ∆x x2 - x1 7 - 1 ∆t t2 - t1 4 - 2 3 * *

Draw a velocity versus time graph for each of the following: constant forward motion constant backward motion constant acceleration constant deceleration sitting still

constant forward motion Velocity stays the same (above 0 m/s) Time (s) Velocity (m/s)

constant backward motion Velocity stays the same (below 0 m/s) Velocity (m/s) Time (s)

constant acceleration Constant upwards slope Velocity at the second point is more than the first Time (s) Velocity (m/s)

constant deceleration Constant downward slope Velocity at the second point is less than the first Time (s) Velocity (m/s)

sitting still Flat line at 0 velocity Time (s) Velocity (m/s)

Velocity -Time Graphs Slope on a velocity time graph is acceleration. Time (s) Velocity (m/s) Rise (ΔY) Run (Δ X) Slope = ___________ Δ Velocity (ΔY) Δ Time (Δ X) Acceleration = _______________ (ΔY) (Δ X) Therefore: slope of V-T graph = acceleration

a = = 2.2 ACCELERATION – The change in velocity over time. In Physics we use the expression: ∆v vf - vi ∆t tf - ti The units for acceleration are usually meters ( m/s/s ) or m/s2 seconds2 a = =

Slope of a velocity vs time = acceleration rise/ run = ∆v/∆t = acceleration

What is the final velocity of a car that accelerates from rest at 4 m/s/s for 3 sec? 2. What is the slope of the line for the red car for the first 3 sec? 3. Does the red car pass the blue car at 3 sec? If not, then when does the red car pass the blue car? 4. When lines on a velocity-time graph intersect, does it mean that the two cars are passing by each other? If not, what does it mean?

1. What is the final velocity of a car that accelerates from rest at 4 m/s/s for three seconds? 12 m/s 2. What is the slope of the line for the red car for the first three seconds? 4 m/s2 3. Does the red car pass the blue car at three seconds? If not, then when does the red car pass the blue car? No, at 9 sec 4. When lines on a velocity-time graph intersect, does it mean that the two cars are passing by each other? If not, what does it mean? No, just same velocity

Displacement and Final Velocity IMPORTANT FORMULAS: Displacement and Final Velocity For an object that accelerates from rest (vi = 0) ∆x = ½ ( vf ) ∆t ( remember: ½ vf = vavg ) vf = a ( ∆t ) ∆x = ½ a( ∆t )2 Vf2 = 2(a)(∆x)