General Physics (PHY 170) Chap 2. Acceleration motion with constant acceleration. Tuesday, January 15, 13

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1 General Physics (PHY 170) Chap 2 Acceleration motion with constant acceleration 1

2 Average Acceleration Changing velocity (non-uniform) means an acceleration is present Average acceleration is the rate of change of the velocity Average acceleration is a vector quantity (i.e. described by both magnitude and direction) 2

3 Average Acceleration When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are opposite, the speed is decreasing Units SI Meters per second squared (m/s 2 ) CGS Centimeters per second squared (cm/s 2 ) US Customary Feet per second squared (ft/s 2 ) 3

4 Acceleration vs. Deceleration Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration: Accelerating Decelerating Decelerating Accelerating 4

5 Instantaneous and Constant Acceleration Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero When the instantaneous accelerations are always the same, the acceleration will be constant (uniform) The instantaneous accelerations will all be equal to the average acceleration 5

6 Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph 6

7 One-dimensional Motion With Constant Acceleration If acceleration is uniform: Shows velocity as a function of acceleration and time: if the acceleration is uniform, then the velocity is changing at a constant rate, it s a straight line with a slope determined by the acceleration. 7

8 One-dimensional Motion with Constant Acceleration Vx V X = Area1 + Area2 = = V o t + (V x -V o )t/2 Vo at Vo V x = V o + at => X = V o t + at 2 /2 0 t t 8

9 Motion with Constant Acceleration The relationship between position and time follows a characteristic curve. Parabola Gives displacement as a function of time, velocity and acceleration 9

10 Summary in the kinematics equations 10

11 Example: An Accelerating Train A train moving in a straight line with an initial velocity of 0.50 m/s accelerates at 2.0 m/s 2 for 2.0 s, coasts with zero acceleration for 3.0 s, and then accelerates at -1.5 m/s 2 for 1.0 s. (a) What is the final velocity v f of the train? (b) What is the average acceleration a av of the train? 11

12 Example: Hit the Brakes! A park ranger driving at 11.4 m/s in back country suddenly sees a deer frozen in the headlights. He applies the brakes and slows with an acceleration of 3.80 m/s 2. (a) If the deer is 20.0 m from the ranger s car when the brakes are applied, how close does the ranger come to hitting the deer? (b) What is the stopping time? 12

Example: A Traveling Electron An electron in a cathode-ray tube accelerates from rest with a constant acceleration of 5.33 x 10 12 m/s 2 for 0.150 s, then drifts with a constant velocity for 0.

13 Example: A Traveling Electron An electron in a cathode-ray tube accelerates from rest with a constant acceleration of 5.33 x m/s 2 for s, then drifts with a constant velocity for s, then slows to a stop with a negative acceleration of 2.67 x m/s 2. (Note: 1 s = 10-6 s). How far does the electron travel? 1) Draw the electron (as a dot). Calculate the displacement x i and velocity v i for each part of the path: 3. The answers have the correct units and appear to be reasonable. 13

14 Free Fall All objects moving under the influence of only gravity are said to be in free fall All objects falling near the earth s surface fall with a constant acceleration This acceleration is called the acceleration due to gravity, and indicated by g 14

15 Free Fall An object falling in air is subject to air resistance (and therefore is not freely falling). 15

16 Acceleration due to Gravity Symbolized by g g = 9.8 m/s² (can use g = 10 m/s² for estimates) g is always directed downward toward the center of the earth The picture shows an apple and a feather falling in vacuum with identical motions. The magnitude of this acceleration, designated as g, has the approximate value of a = g = 9.81 m/s 2 = 32.2 ft/s 2. If downward is designated as the +y direction, then a = +g; if downward is designated as the y direction, then a = g. (Note that g is always positive., but a may have either sign.) 16

17 Freely Falling Objects Free fall: use the kinematic equations Y Generally use y instead of x since vertical 17

18 Free Fall -- an Object Dropped y Initial velocity is zero Frame: let up be positive -g v o = 0 a = -g x 18

19 Free Fall -- an Object Thrown Downward With upward being positive, acceleration will be negative, g = 9.8 m/s² y a = -g Vo x Initial velocity 0 -g With upward being positive, initial velocity will be negative 19

20 Problem Solving Strategy for 1-D Motion with Constant Acceleration Picture - Determine if the problem is asking you to find time, distance, velocity, or acceleration for an object. Solve - Use the following steps to solve problems that involve onedimensional motion and constant acceleration: Draw a figure showing the particle in its initial and final positions. Include a coordinate axis and label initial and final coordinates of the position. Select one or more of the constant-acceleration kinematic equations. Solve them algebraically for the desired quantities. Substitute in the given values and evaluate the answer. Repeat as needed. Check - Make sure your answers are dimensionally consistent. Make sure the magnitudes of your answers are in the expected ballpark. 20

21 Free Fall -- object thrown upward V = 0 Initial velocity is upward, so positive -g The instantaneous velocity at the maximum height is zero Vo a = g everywhere in the motion g is always downward, a is negative 21

22 Freely Falling Objects - Trajectory of a projectile The motion may be symmetrical then t up = t down Position then v f = -v o Velocity Acceleration 22

23 Example:Speed of a Lava Bomb A volcano shoots out blobs of molten lava (lava bombs) from its summit. A geologist observing the eruption uses a stopwatch to time the flight of a particular lava bomb that is projected straight upward. If the time for it to rise and fall back to its launch height is 4.75 s, What is its initial speed and how high did it go? (Use g = 9.81 m/s 2.) 3. The answers have the correct units and appear to be reasonable. 23

Example: The Flying Cap Upon graduation, a joyful student throws her cap straight up in the air with an initial speed of 14.7 m/s. Given that its acceleration has a magnitude of 9.

24 Example: The Flying Cap Upon graduation, a joyful student throws her cap straight up in the air with an initial speed of 14.7 m/s. Given that its acceleration has a magnitude of 9.81 m/s 2 and is directed downward (we neglect air resistance), (a) When does the cap reach its highest point? (b) What is the distance to the highest point? (c) Assuming the cap is caught at the same height it was released, what is the total time that the cap is in flight? Draw the cap (as a dot) in its various positions (a) Use the time, velocity and acceleration relation. (b) y = Vt -(1/2 gt 2 ) = 0 - (-11.0) = 11.0 m (c) Up time = down time, so total time is 3.0 s. 24

Example: The Flying Cap (continued) The height of the cap vs. time has the form of a parabola (since y ~ t 2 ). It is symmetric about the midpoint (without air resistance).

25 Example: The Flying Cap (continued) The height of the cap vs. time has the form of a parabola (since y ~ t 2 ). It is symmetric about the midpoint (without air resistance). The velocity of the cap vs. time has the form of a straight line (since v ~ t). The velocity crosses zero at the midpoint and is negative thereafter, because the cap is moving downward. 25

26 Non-symmetrical Free Fall The motion may not be symmetrical Break the motion into various parts Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the non-symmetrical portion 26

Be on time Switch off mobile phones. Put away laptops. Being present = Participating actively

A couple of house rules Be on time Switch off mobile phones Put away laptops Being present = Participating actively http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html Het basisvak Toegepaste Natuurwetenschappen