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Understanding Motion Along a Straight Line: Velocity and Acceleration Concepts

This review explores the fundamental concepts of motion along a straight line, focusing on velocity and acceleration. By defining key terms such as average and instantaneous velocity, and elaborating on their relation to time and displacement, we facilitate a deeper understanding of motion. The relationship between velocity and acceleration is illustrated through the use of derivatives, with graphical representations on x-t and v-t graphs. Key equations of motion under constant acceleration, including those for freely falling objects, are presented, guiding you to apply differentiation and integration to solve motion problems accurately.

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Understanding Motion Along a Straight Line: Velocity and Acceleration Concepts

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

  2. velocity Average velocity You can choose the origin, where x = 0, and the (+)-vex-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

  3. Velocity on x-t graph Smaller v Larger v

  4. acceleration Velocity: the rate of change of position with timeAcceleration: 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.

  5. acceleration on x-t graph Curvature downward Curvature upward c is the second derivative of the curve at t=0. Thus from the x-t graphyou 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 v=0 since dx/dt =0 a<0 region

  6. Velocity and position by integration You can also obtain v(t) and x(t), when a(t) is known/given. differentiation differentiation integration integration First obtain v(t) from a(t) then obtain x(t) from v(t). You can set t0 = 0

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

  8. 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.

  9. 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. initial velocity Choose the origin acceleration due to gravitymagnitude: g = 9.8 m/sdirection: downward always true ground

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

  11. 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 v0, H, g, etc. • If you have to give numerical answers, be careful with the unit.

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