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Chapter 2: Kinematics in one Dimension Displacement Velocity Acceleration HW2: Chap. 2: pb.3,pb.8,pb.12,pb.22,pb.27,pb.29,pb.46

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Position on a line 1.Reference point (origin) 2.Distance 3.Direction Symbol for position: x SI units: meters, m

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Displacement on a line xfxf xixi Change of position is called Displacement: Displacement is a vector quantity It has magnitude and direction

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Displacement Defined as the change in position during some time interval Represented as x SI units are meters (m) x can be positive or negative Different than distance – the length of a path followed by a particle. Displacement has both a magnitude and a direction so it is a vector.

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Reference Frames and Displacement We make a distinction between distance and displacement. Displacement (blue line) is how far the object is from its starting point, regardless of how it got there. Distance traveled (dashed line) is measured along the actual path.

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Reference Frames and Displacement The displacement is written: Left: Displacement is positive. Right: Displacement is negative.

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Vectors and Scalars Vector quantities need both magnitude (size or numerical value) and direction to completely describe them Will use + and – signs to indicate vector directions Scalar quantities are completely described by magnitude only

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Average Speed and Average Velocity Speed is how far an object travels in a given time interval: Velocity includes directional information:

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Average Speed Average speed =distance traveled/ time elapsed Example: if a car travels 300 kilometer (km) in 2 hours (h), its average speed is 150km/h. Not to confuse with average velocity.

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Example

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Average Velocity The average velocity is rate at which the displacement occurs The SI units are m/s Is also the slope of the line in the position – time graph

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Average Velocity, cont Gives no details about the motion Gives the result of the motion It can be positive or negative It depends on the sign of the displacement It can be interpreted graphically It will be the slope of the position-time graph

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Example Mary 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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Mary walked a distance of 12 meters in 24 seconds; thus, her average speed was 0.50 m/s.distance However, since her displacement is 0 meters, her 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.displacement

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Not to Confuse Speed is a number : a scalar Velocity is a vector : with magnitude and direction

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Example 2-1: Runner’s average velocity. The position of a runner as a function of time is plotted as moving along the x axis of a coordinate system. During a 3.00-s time interval, the runner’s position changes from x 1 = 50.0 m to x 2 = 30.5 m, as shown. What was the runner’s average velocity? Average Velocity

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x2x2 t2t2 Average velocity from a graph of x(t) Time (t) Position (x) x1x1 t1t1 v(t) = slope of x(t)

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Average Velocity Example 2-2: Distance a cyclist travels. How far can a cyclist travel in 2.5 h along a straight road if her average velocity is 18 km/h?

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Instantaneous Velocity The instantaneous velocity is the average velocity in the limit as the time interval becomes infinitesimally short. Ideally, a speedometer would measure instantaneous velocity; in fact, it measures average velocity, but over a very short time interval.

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Instantaneous velocity from a graph of x(t) x t Time (t) Position (x) tt v(t) = slope of x(t) +direction -direction Sign + slope - slope P1P1 P2P2

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Instantaneous Velocity The instantaneous speed always equals the magnitude of the instantaneous velocity; it only equals the average velocity if the velocity is constant.

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Instantaneous Velocity Example 2-3: Given x as a function of t. A jet engine moves along an experimental track (which we call the x axis) as shown. We will treat the engine as if it were a particle. Its position as a function of time is given by the equation x = At 2 + B, where A = 2.10 m/s 2 and B = 2.80 m. (a) Determine the displacement of the engine during the time interval from t 1 = 3.00 s to t 2 = 5.00 s. (b) Determine the average velocity during this time interval. (c) Determine the magnitude of the instantaneous velocity at t = 500 s.

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Acceleration Acceleration is the rate of change of velocity. Example 2-4: Average acceleration. A car accelerates along a straight road from rest to 90 km/h in 5.0 s. What is the magnitude of its average acceleration?

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Acceleration Conceptual Example 2-5: Velocity and acceleration. (a) If the velocity of an object is zero, does it mean that the acceleration is zero? (b) If the acceleration is zero, does it mean that the velocity is zero? Think of some examples.

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Acceleration Example 2-6: Car slowing down. An automobile is moving to the right along a straight highway, which we choose to be the positive x axis. Then the driver puts on the brakes. If the initial velocity (when the driver hits the brakes) is v 1 = 15.0 m/s, and it takes 5.0 s to slow down to v 2 = 5.0 m/s, what was the car’s average acceleration ?

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Acceleration There is a difference between negative acceleration and deceleration: Negative acceleration is acceleration in the negative direction as defined by the coordinate system. Deceleration occurs when the acceleration is opposite in direction to the velocity.

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