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Vertical Circular Motion Test
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Calculus in 2D Know how to apply variable acceleration with vectors Understand how to apply core 3 calculus in questions Be able to find a direction and magnitude
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Recap DisplacementVelocityAcceleration You should recall from last lesson the how displacement, velocity and acceleration can be linked by differentiation and integration, if we are given an equation for one of them in terms of t. Don’t forget to consider the constant of integration +c when you are integration. You can use one set of values you know to work this out.
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Var. Acc. in 2D In 2D we will consider having an equation for the position vector that describes the motion. r = xi + yj x and y can be a function of t to describe a particle in motion. We can just differentiate each component separately to find v and a. O i j x y
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Magnitude and Direction Can be calculated using Pythagoras. Direction can be calculated using trigonometry. The magnitude of r is denoted by|r| or r. Similarly |v| or |a| can be used. O i j 4 2 r = 4i + 2j θ
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Other ideas about direction We will try a couple of examples to deal with a few more things about direction. Such as : – finding when something is moving in the direction of i. – Finding when something is travelling North East (if i is defined as east and j is defined as north)
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Example 1 The position vector of a particle is given by r = sin2ti +(t + cos 2t)j. a)Find an expression for the velocity of the particle at time t. b)Find the first value of t after t=0 for which the particle is moving in the direction of i. c)Show that the magnitude of the acceleration of the particle is constant.
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Example 2 The position vector rkm, at time t hours of a boat relative to point O is given by r = (1 – cos3t)i – 2tj. Where i and j are unit vectors in the directions east and north respectively. Find the value of t for which the boat is moving south-east.
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North (j) East (i) South East (i-j) Velocity is a multiple of (i-j)
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Integration Example Find an expression for v and r in terms of t, given that a = (e -t )i + (1 – e -2t )j and v = 0 and r = 0 when t = 0
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Question Time Try out the exam questions
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Independent Study Differentiation Exercise B p 52 Q 1, 3, 5, 7 (solutions p 154) Integration Exercise C p56 Q 2, 3, 4 (solutions p 156)
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Independent Study Try out the rest of exercise B/C p52-56
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