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PHYS 218 sec. 517-520 Review Chap. 2 Motion along a straight line

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velocity Average velocity You can choose the origin, where x = 0, and the (+)-ve x-direction for convenience. Once you fix them, keep this convention. Instantaneous velocity = velocity Velocity at any specific instant of time or specific point along the path Definition of the derivative

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Velocity on x-t graph Larger v Smaller v

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acceleration Velocity: the rate of change of position with time Acceleration: the rate of change of velocity with time Average acceleration (Instantaneous) acceleration acceleration on v-t graph Acceleration is to velocity as velocity is to position. Therefore, in v-t graph, the slope of tangent line of v(t) at a given t is the acceleration at that time.

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acceleration on x-t graph Curvature upward Curvature downward c is the second derivative of the curve at t=0. Thus from the x-t graph you can know the acceleration qualitatively even though you cannot not know its magnitude. In an x-t graph, the slope of the curve gives the velocity, while the curvature gives the sign of the acceleration. a<0 region a>0 region v=0 since dx/dt =0

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Velocity and position by integration You can also obtain v(t) and x(t), when a(t) is known/given. differentiation integration First obtain v(t) from a(t) then obtain x(t) from v(t). You can set t 0 = 0

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Constant acceleration If a=constant, you can easily calculate the integrals. When a = constant This gives the familiar expressions for constant acceleration.

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Some relations can be obtained for 1-dim. motion with a constant acceleration. Here we eliminate t! This give a relation between v, a and x. Here we eliminate a! This give a relation between v, t and x. Do not try to memorize these formulas. If you understand the equation of motion, these relations follow in a natural way.

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Freely falling objects Typical example of 1-dim. motion Choose the upward as the (+)-ve y-direction This is a convention. You can make other choice. Choose the origin initial velocity acceleration due to gravity magnitude: g = 9.8 m/s direction: downward ground always true

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Freely falling objects (2) Maximum height h Time when it hits the ground Since you know the time t H, you can know its velocity at t H.

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Tips Can you obtain v(t) and x(t) if a(t) is given? To do this, you should be familiar with differentiation and integration. Find the proper mathematical equation to describe the motion, i.e. formulate the situation. Here are some examples. What equation describes the maximum height? The ball hits the ground, how do you describe this situation in a mathematical formula? Give your answer in terms of the given information such as v 0, H, g, etc. If you have to give numerical answers, be careful with the unit.

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2.2 The Derivative as a Function. 22 We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the.

2.2 The Derivative as a Function. 22 We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the.

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