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Chapter 2: One-Dimensional Motion
- Motion at fixed velocity
- Definition of average velocity
- Motion with fixed acceleration
- Graphical representations 1 Displacement vs. position Position: x (relative to origin) Displacement: !x = xf-xi 2 basic formula v =! x ! t = x (^) f " xi t Average velocity Average velocity
- Can be positive or negative
- Depends only on initial/final positions
- e.g., if you return to original position, average velocity is zero 3 basic formula v = ! x ! t
x (^) f " xi t Instantaneous velocity Let time interval approach zero
- Defined for every instance in time
- Equals average velocity if v = constant
- SPEED is absolute value of velocity 4 Graphical Representation of Average Velocity Between A and D , v is slope of blue line 5 Graphical Representation of Instantaneous Velocity v(t=3.0) is slope of tangent (green line) 6
Example 2. Carol starts at a position x(t=0) = 1.5 m. At t=2.0 s, Carol’s position is x(t=2 s)=4.5 m At t=4.0 s, Carol’s position is x(t=4 s)=-2.5 m a) What is Carol’s average velocity between t=0 and t=2 s? b) What is Carol’s average velocity between t=2 and t=4 s? c) What is Carol’s average velocity between t=0 and t=4 s? a) 1.5 m/s b) -3.5 m/s c) -1.0 m/s 7 Example 2. On a mission to rid Spartan Stadium of vermin, an archer shoots an arrow across the stadium at an unlucky rat 200 meters away. The archer hears the squeal 2. seconds later. What was the velocity of the arrow? The speed of sound is 330 m/s. 8 Example 2.2: Visualize the problem! 9 Example 2. On a mission to rid Spartan Stadium of vermin, an archer shoots an arrow across the stadium at an unlucky rat 200 meters away. The archer hears the squeal 2. seconds later. What was the velocity of the arrow? The speed of sound is 330 m/s. V = 125 m/s 10 Example 2.3a The instantaneous velocity is zero at ___ A) a B) b & d C) c & e 11 Example 2.3b The instantaneous velocity is negative at _____ A) a B) b C) c D) d E) e 12
Graphical Description of Acceleration Acceleration is slope of tangent line in v vs. t graph 19 Graphical Description of Acceleration a > 0 a < 0 a is positive/negative when v vs. t is rising/falling or when x vs t curves upwards/downwards 20 Example a b c d e At which point(s) does the position equal zero? A) a only B) a and d C) b only D) b & d 21 Example a b c d e At which point(s) does the velocity equal zero? A) a B) b only C) c only D) b & d E) a & d 22 Example 2.5c a b c d e At which point is the velocity negative? A) a B) b C) c D) d E) e 23 Example a b c d e At which segment(s) is the acceleration negative? A) a-c B) c-d C) c-e D) d-e 24
Example 2.5e a b c d e At which point(s) does the acceleration equal zero? A) none of the below B) b C) c D) d E) e 25 Constant Acceleration
- a vs. t is a constant
- v vs t is a straight line
- x vs t is a parabola Eq.s of Motion v (^) f = v 0 + at ! x =
( v 0 + v (^) f ) t 26 Solving Problems with Eq.s of Motion 5 variables: !x, t, v 0 , vf, a 2 equations: v (^) f = v 0 + at ! x =
( v 0 + v (^) f ) t 3 variables must be given so that 2 equations can solve for 2 unknowns 27 Example 2. Crash Houlihan speeds down the intersate, when she slams on the brakes and slides into a concrete barrier. The police measure skid marks to be 60 m long, and from a tape recording, know that she was breaking from 3.5 seconds. Furthermore, they know that her Mercedes would decelerate at 5.5 m/s^2 while skidding. What was Crash’s speed when she hit the barrier? 7.52 m/s 28 Other Forms of Eq.s of Motion ! x = v 0 + ( v 0 + at ) 2 t ! x = v 0 t +
at^2 Substitute to eliminate vf v (^) f = v 0 + at ! x =
( v 0 + v (^) f ) t 29 Other Forms of Eq.s of Motion ! x = ( v (^) f " at ) + v (^) f 2 t ! x = v (^) f t "
at^2 Substitute to eliminate v 0 v (^) f = v 0 + at ! x =
( v 0 + v (^) f ) t 30
Free Fall
- Objects under the influence of gravity (no resistance) fall with constant downward acceleration (if near Earth’s surface). g = 9.81 m/s^2
- Use the usual equations with a --> -g 37 - Father was a musician, experimented with music - Initially was a professor teaching pre-meds - Developed telescope ~ 1610: Milky Way = stars Moons of Jupiter Phases of Venus… - Measured g - Quantified mechanics - In 1632, published Dialogue concerning the two greatest world systems - Was found guilty of heresy Galileo 38 Example 2.8a A man drops a brick off the top of a 50-m building. The brick has zero initial velocity. a) How much time is required for the brick to hit the ground? b) What is the velocity of the brick when it hits the ground? A B c a) 3.19 s b) -31.3 m/s 39 Example 2.8b A man throws a brick upward from the top of a 50 m building. The brick has an initial upward velocity of 20 m/s. a) How high above the building does the brick get before it falls? b) How much time does the brick spend going upwards? c) What is the velocity of the brick when it passes the man going downwards? d) What is the velocity of the brick when it hits the ground? e) At what time does the brick hit the ground? A B c 40 Example 2.8b A man throws a brick upward from the top of a 50 m building. The brick has an initial upward velocity of 20 m/s. a) How high above the building does the brick get before it falls? b) How much time does the brick spend going upwards? c) What is the velocity of the brick when it passes the man going downwards? d) What is the velocity of the brick when it hits the ground? e) At what time does the brick hit the ground? a) 20.4 m b) 2.04 s c) -20 m/s d) -37.2 m/s e) 5.83 s 41 Example 2.9a A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) At ‘A’ the acceleration is positive A C D A C D B E a) True b) False 42
Example 2.9b A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) At ‘B’ the velocity is zero A C D A C D B E a) True b) False 43 Example 2.9c A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) At ‘B’ the acceleration is zero A C D A C D B E a) True b) False 44 Example 2.9d A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) At ‘C’ the velocity is negative A C D A C D B E a) True b) False 45 Example 2.9e A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) At ‘C’ the acceleration is negative A C D A C D B E a) True b) False 46 Example 2.9f A man throws a brick upward from the top of a building. (Assume the coordinate system is defined with positve defined as upward) The speed at ‘C’ and at ‘A’ are equal A C D A C D B E a) True b) False 47 Example 2.9g A man throws a brick upward from the top of a building. TRUE OR FALSE. (Assume the coordinate system is defined with positve defined as upward) The velocity at ‘C’ and at ‘A’ are equal A C D A C D B E a) True b) False 48
