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KINEMATICS Speed and Velocity

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Speed and Velocity Just as distance and displacement have distinctly different meanings (despite their similarities), so do speed and velocity.

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Speed Speed is a scalar quantity which refers to "how fast an object is moving." A fast-moving object has a high speed while a slow-moving object has a low speed. An object with no movement at all has a zero speed.

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Velocity Velocity is a vector quantity which refers to "the rate at which an object changes its position."

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Velocity Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. While this might result in a frenzy of activity, it would result in a zero velocity. Because the person always returns to the original position, the motion would never result in a change in position. Since velocity is defined as the rate at which the position changes, this motion results in zero velocity.

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Recap Speed is a scalar and does not keep track of direction; velocity is a vector and is direction- aware.

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Direction The task of describing the direction of the velocity vector is easy! The direction of the velocity vector is simply the same as the direction in which an object is moving.

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Direction It would not matter whether the object is speeding up or slowing down, if the object is moving rightwards, then its velocity is described as being rightwards. If an object is moving downwards, then its velocity is described as being downwards.

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**Average Speed As an object moves, it often undergoes changes in speed.**

One instant, the car may be moving at 50 mi/hr and another instant, it might be stopped (i.e., 0 mi/hr). Yet during the course of the trip to school the person might average 25 mi/hr.

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Special Note It is important to realize that the average speed for a trip cannot, in most cases, be calculated by finding the speed for individual stages and dividing by the number of stages.

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Formula The average speed during the course of a motion is often computed using the following equation:

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**Instantaneous vs Average Speed**

Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows.

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**Instantaneous vs Average Speed**

Instantaneous Speed - speed at any given instant in time. Average Speed - this is the constant speed that would be need to cover the same distance in the same amount of time; found simply by a distance/time ratio.

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**Instantaneous vs Average Speed**

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Average Velocity Meanwhile, the average velocity is often computed using the equation

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**Instantaneous vs. Average Velocity**

Instantaneous Velocity - instantaneous speed plus a direction. Consider a Ferris wheel that is moving at a constant speed. A rider at the top has the same instantaneous speed as a rider at the bottom, but the instantaneous velocities are in opposite directions. Average Velocity – For example, when a car moved 50 km west in 2 hours, the average velocity is 25 km/h west because 50km/2h = 25 km/h west.

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Examples Example 1 While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours. What was her average speed?

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Solution To compute her average speed, we simply divide the distance of travel by the time of travel.

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Example 2 Now let's try a little more difficult case by considering the motion of that physics teacher again. The physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and the average velocity.

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Solution 2 The physics teacher walked a distance of 12 meters in 24 seconds; thus, his average speed was 0.50 m/s. However, since her displacement is 0 meters, his average velocity is 0 m/s. Remember that the displacement refers to the change in position and the velocity is based upon this position change. In this case of the teacher's motion, there is a position change of 0 meters and thus an average velocity of 0 m/s.

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Example 3 The diagram below shows the position of a cross-country skier at various times. At each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from A to B to C to D. Use the diagram to determine the average speed and the average velocity of the skier during these three minutes.

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**Solution 3 Average speed = total distance/total time = 420m / 3 min**

Average velocity = displacement/time = 140 m / 3 min = 46.7 m/min

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**Example 4 Seymour Butz views football games from under the bleachers.**

What is Seymour's average speed and average velocity?

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**Solution 4 Average speed = 95 yards/10 min = 9.5 yards/min**

Average velocity = 55 yards / 10 min = 5.5 yards/min

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Conclusion In conclusion, speed and velocity are kinematic quantities which have distinctly different definitions. Speed, being a scalar quantity, is the distance (a scalar quantity) per time ratio. Speed is ignorant of direction. On the other hand, velocity is direction- aware. Velocity, the vector quantity, is the rate at which the position changes. It is the displacement or position change (a vector quantity) per time ratio.

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Constant Speed If a speed is constant, then the distance traveled every second is the same. An object with a changing speed would be moving a different distance each second.

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Constant Velocity An object with constant velocity has a constant speed in a constant direction. The object must be traveling in a straight line. Constant velocity is also called Uniform Motion.

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Example

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Example 4 A bus moving at a constant speed for one hour travels 100 km. It moves at a different constant speed for the next two hours and travels 140 km. What is its average speed for the total trip?

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**Solution 4 CORRECT V = distance/time d=100 km+140 km = 240 km.**

t=1 h+2h = 3h. 240km / 3h = 80 km/h INCORRECT Vav1 = 100km/1h = 100 km/h Vav2 = 140km/2h = 70 km/h V = [100 km/h +70 km/h] / 2 = 85 km/h

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POSITION-TIME GRAPHS

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