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Unformatted text preview: John Ellison, UCR p.1 Notes on Chapter 3 Velocity and Acceleration in Three Dimensions The position of a particle can be described by a position vector , which is a vector extending from the origin to the particle. We can write it in component form as where x , y , and z are the components of . As the particle moves, moves in such a way that it always points from the origin to the particle, so is a function of time. The average velocity v avg is defined by where is the displacement during the time interval ∆ t . Note that in the last step we have written the displacement vector in terms of its vector components. The instantaneous velocity (or simply velocity ) v is defined by The velocity vector is tangent to the path of the particle at time t . The (instantaneous) speed s of the particle is just the magnitude of the velocity v . Example The plot of x versus y below shows the path of a particle. The displacement over the time interval from 10 s to 25 s is shown as the vector . The average velocity has a magnitude of ∆ r / (25 s - 10 s) and points in the same direction as . The (instantaneous) velocity at time t = 10 s points in the direction of the tangent to the path of the particle at t = 10 s. It is indicated by the red vector. r = x ˆ i A y ˆ j A z ˆ k r 1 r 2 ∆ r v avg = displacement time interval = ∆ r ∆ t = ∆ x ˆ i A ∆ y ˆ j A ∆ z ˆ k ∆ t ∆ r = r 2 B r 1 v = = d r d t = dx dt ˆ i A dy dt ˆ j A dz dt ˆ k v r r r ∆ r ∆ r John Ellison, UCR p.2 When a particle’s velocity changes from to in a time interval ∆ t , its average acceleration a avg during is The instantaneous acceleration (or acceleration ) at time t is defined by If the velocity changes in either magnitude or direction , the particle has an acceleration. Example Particle A moves along the line y = 30 m with a constant velocity of magnitude of 3.0 m/s and directed parallel to the positive x axis. Particle B starts at the origin with zero speed and constant acceleration (of magnitude 0.40 m/s 2 ) at the same instant that particle A passes the y axis. What angle θ between and the +y axis would result in a collision between these two particles? Solution The situation is shown in the figure at right. If the two particles are to collide they must have the same position vector at some instant of time. So we need to express the position vectors in terms of time t and angle θ . Since A has constant velocity we can use x = x + vt , but this only applies to the motion parallel to the x-direction. There is no motion along the y-direction, so y = constant. With x = 0 we get Since B has constant acceleration we can use x- x = v t + ½ at 2 ....
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- Winter '08
- Acceleration, Velocity, John Ellison