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Transcript
M2 – CHAPTER 1 – MOTION IN A PLANE In one-dimensional motion, you differentiate displacement with respect to time to obtain velocity and then differentiate velocity with respect to time to obtain acceleration, and integrate to go the opposite way. For a particle moving in a plane the same relationships apply, but the position, velocity and acceleration are expressed as vectors. To differentiate a vector you need to differentiate each of its components. Displacement: Velocity: Acceleration: 𝐫 = 𝑥𝐢 + 𝑦𝐣 𝐫̇ = 𝑥̇ 𝐢 + 𝑦̇ 𝐣 𝐫̈ = 𝑥̈ 𝐢 + 𝑦̈ 𝐣 𝐫̇ = dr dt and 𝐫̈ = d2 r dt2 The components must be functions of time. To emphasise this you sometimes write r = f(t)i + g(t)j When integrating, this is done in the same way as you would normally, however you must include an arbitrary constant for both i and j components. So: 𝐯 = ∫ a dt + C𝐢 + D𝐣 Example 1 The displacement, in m, of a particle at time t seconds is given by r = 4t2i + (3t – 5t3)j. Find: (a) its speed when t = 1 (b) the direction in which it is accelerating at that time. Example 2 The acceleration of a particle at time t is a = 4i + 6tj. Find an expression for its velocity. Example 3 The velocity of a particle at time t is given by v = 6t2i + (3t2 – 8t)j. Find its position vector at time t, given that r = 5i – 6j when t = 0. Example 4 A particle moves in a plane with an acceleration 2tj ms-2. At time t = 0 the particle is at the point (i + 4j) m and has velocity (3i – 4j) ms-1. Find expressions for (a) the velocity of the particle (b) the position of the particle at time t. Example 5 A particle of mass 4kg is acted on by a force (8i + 12tj) N. Initially the particle has velocity (3i – 5j) ms-1. Find its velocity after 4 seconds. Example 6 A particle of mass 3kg moves under the action of a force of (6i – 36t2j) N. The particle is initially at rest at the point with position vector (i + j) m. (a) Find the velocity and position of the particle at time t. (b) If the position of the particle at time t is (xi + yj)m, find an equation connecting x and y, and deduce that the path of the particle is a parabola. EXERCISE
