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**Motion in One Dimension**

Chapter 2 Motion in One Dimension

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Dynamics The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts Kinematics is a part of dynamics In kinematics, you are interested in the description of motion Not concerned with the cause of the motion

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**Quantities in Motion Any motion involves three concepts**

Displacement Velocity Acceleration These concepts can be used to study objects in motion

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**Position Defined in terms of a frame of reference**

One dimensional, so generally the x- or y-axis Defines a starting point for the motion

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**Displacement Defined as the change in position**

f stands for final and i stands for initial May be represented as y if vertical Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

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Displacements

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**Vector and Scalar Quantities**

Vector quantities need both magnitude (size) and direction to completely describe them Generally denoted by boldfaced type and an arrow over the letter + or – sign is sufficient for this chapter Scalar quantities are completely described by magnitude only

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**Displacement Isn’t Distance**

The displacement of an object is not the same as the distance it travels Example: Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zero

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Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Speed is a scalar quantity

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Speed, cont Average speed totally ignores any variations in the object’s actual motion during the trip The total distance and the total time are all that is important SI units are m/s

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**Velocity It takes time for an object to undergo a displacement**

The average velocity is rate at which the displacement occurs generally use a time interval, so let ti = 0

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Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Other units may be given in a problem, but generally will need to be converted to these

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Speed vs. Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger

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**Graphical Interpretation of Velocity**

Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line

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**Average Velocity, Constant**

The straight line indicates constant velocity The slope of the line is the value of the average velocity

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**Average Velocity, Non Constant**

The motion is non-constant velocity The average velocity is the slope of the blue line joining two points

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**Instantaneous Velocity**

The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero The instantaneous velocity indicates what is happening at every point of time

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**Instantaneous Velocity on a Graph**

The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocity

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**Uniform Velocity Uniform velocity is constant velocity**

The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity

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Acceleration Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

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**Average Acceleration Vector quantity**

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 in the opposite directions, the speed is decreasing

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**Instantaneous and Uniform Acceleration**

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 uniform The instantaneous accelerations will all be equal to the average acceleration

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**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

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Average Acceleration

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**Relationship Between Acceleration and Velocity**

Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero

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**Relationship Between Velocity and Acceleration**

Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration

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**Relationship Between Velocity and Acceleration**

Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative

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Kinematic Equations Used in situations with uniform acceleration

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Notes on the equations Gives displacement as a function of velocity and time Use when you don’t know and aren’t asked for the acceleration

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Notes on the equations Shows velocity as a function of acceleration and time Use when you don’t know and aren’t asked to find the displacement

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**Graphical Interpretation of the Equation**

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Notes on the equations Gives displacement as a function of time, velocity and acceleration Use when you don’t know and aren’t asked to find the final velocity

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Notes on the equations Gives velocity as a function of acceleration and displacement Use when you don’t know and aren’t asked for the time

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**Problem-Solving Hints**

Read the problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Label all quantities, be sure all the units are consistent Convert if necessary Choose the appropriate kinematic equation

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**Problem-Solving Hints, cont**

Solve for the unknowns You may have to solve two equations for two unknowns Check your results Estimate and compare Check units

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Example 1 A bicyclist rides her bike at a speed of 10 Km/hr for 25 min, remembers that she forgot something at home she turns around and pedals at a rate of 12Km/hr back to her house. If turning around time is ignored what are the following? The distance she was away from home when she decided to turn around. The time that it took her to return home after she remembered that she forgot something. The average speed of the entire trip. The average velocity of the entire trip.

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Example 2 A motorist drives south for 45.0 minutes at 75.0 km/h and then stops for 20.0 minutes. He then continues south, traveling 120 km in 2.50 h. (a) What is his total displacement? (b) What is his average velocity?

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Example 3 A certain horse ran a race with the following times: 1st quarter 24.5s, 2nd quarter 23.5s, 3rd quarter 22.4s and the 4th quarter time was 24.2s. (a) Find the average speed during each quarter-mile segment. (b) Assuming that the horse’s instantaneous speed at the finish line was the same as the average speed during the final quarter mile, find the average acceleration for the entire race. (Hint: Recall that horses in the race start from rest.)

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Example 4 A certain car is capable of accelerating at a rate of m/s2. How long does it take for this car to go from a speed of 45 mi/h to a speed of 60 mi/h?

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Example 5 Two cars are traveling along a straight line in the same direction, the lead car at 35.0 m/s and the other car at 45.0 m/s. At the moment the cars are 50.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –1.50 m/s2. (a) How long does it take for the lead car to stop? (b) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car? (c) How long does it take for the chasing car to stop?

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Example 6 A drag racer starts her car from rest and accelerates at m/s2 for a distance of 440 m. (a) How long did it take the race car to travel this distance? (b) What is the speed of the race car at the end of the run?

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Example 7 A car traveling in a straight line has a velocity of +5.0 m/s at some instant. After 4.0 s, its velocity is +8.0 m/s. What is the car’s average acceleration during the 4.0-s time interval?

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Example 8 a) What is the average velocity of the object during the 0 to 4s time interval? b) What is the instantaneous velocity of the object at point A? c) What is the average velocity of the object during the 8 to 12s time interval?

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Galileo Galilei Galileo formulated the laws that govern the motion of objects in free fall Also looked at: Inclined planes Relative motion Thermometers Pendulum

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Free Fall All objects moving under the influence of gravity only are said to be in free fall Free fall does not depend on the object’s original motion All objects falling near the earth’s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g

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Elephant and Feather

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**Acceleration due to Gravity**

Symbolized by g g = m/s² When estimating, use g » -10 m/s2 g is always directed downward toward the center of the earth Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion

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**Free Fall – an object dropped**

Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is g = -9.80 m/s2 vo= 0 a = g

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Example 9 A rock is dropped from the top of a cliff and lands at the base of the cliff 3.2s after it is released. How far did the rock drop?

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Example 10 Another scheme to catch the roadrunner has failed! Now a safe falls from rest from the top of a 25.0-m-high cliff toward Wile E. Coyote, who is standing at the base. Wile first notices the safe after it has fallen 15.0 m. How long does he have to get out of the way?

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Example 11 A stunt man sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 10.0 m/s, and the man is initially 3.00 m above the level of the saddle. (a) What must be the horizontal distance between the saddle and the limb when the man makes his move? (b) How long is he in the air?

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**Free Fall – an object thrown downward**

a = g = m/s2 Initial velocity 0 With upward being positive, initial velocity will be negative

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Example 13 A student standing on the top of the science building on campus throws a water balloon at his physics instructor. If the initial velocity of the balloon is -2.3m/s, and it just misses the instructor and lands on the ground in front of him 1.2s after it was released from what height was the balloon released? How fast was it moving when it struck the ground?

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**Free Fall -- object thrown upward**

Initial velocity is upward, so positive The instantaneous velocity at the maximum height is zero a = g = m/s2 everywhere in the motion v = 0

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**Thrown upward, cont. The motion may be symmetrical**

Then tup = tdown Then v = -vo The motion may not be symmetrical Break the motion into various parts Generally up and down

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Example 14 It is possible to shoot an arrow at a speed as high as 100 m/s. (a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up? (b) How long would the arrow be in the air?

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Example 15 A ball is thrown vertically upward with a speed of 35.0 m/s. (a) Where is the ball 4.00 seconds after it was released? (b) What is its velocity at the time? (c) Has it reached its maximum height yet? Defend your answer.

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**Non-symmetrical Free Fall**

Need to divide the motion into segments Possibilities include Upward and downward portions The symmetrical portion back to the release point and then the non-symmetrical portion

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Example 16 A ball is thrown upward with a velocity of +4m/s from the edge of a 10m building. a) How long does it take the ball to reach its maximum height? b) What is the maximum height of the ball? c) If the ball just misses the edge of the building on the way back down how long will the ball be in the air? d) How fast is the ball moving when it strikes the ground?

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Combination Motions

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Example 17 A model rocket is launched straight upward with an initial speed of 50.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s2 until its engines stop at an altitude of 150 m. (a) What is the maximum height reached by the rocket? (b) How long after lift-off does the rocket reach its maximum height? (c) How long is the rocket in the air?

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