Unit Kinematics Module Investigating One Dimensional Motion page 1 of 2 Instantaneous Velocity and the Derivative The instantaneous velocity is the derivative with respect to time of the position x dx function v lim t 0 t dt The instantaneous velocity of a position function of the form x at n is given by dx nat n 1 the equation v dt The instantaneous velocity is the slope of the tangent to the position function The idea of instantaneous velocity comes from finding the average velocity over smaller and smaller time intervals In the limit as the time interval approaches zero the instantaneous velocity is x dx v lim On the graph the instantaneous t 0 t dt velocity is the line that is tangent to the position curve The position function of the iguanodon is a straight line through the origin as shown on the graph The equation for the iguanodon s motion is x ct where c is a constant The instantaneous velocity is found by taking the derivative of the position function with respect to dx d ct 1 1 time v 1 ct c The dt dt iguanodon s velocity is constant since his motion is uniform A ball tossed in the air does not have uniform motion as shown by the parabolic position curve One way to find the instantaneous velocity of the ball at for example t 1 is to draw a line tangent to the curve straight line and measure its slope This method is somewhat crude in that it depends on the accuracy of your drawing and your ability to measure the slope precisely www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1774 rev 03 22 2001 Unit Kinematics Module Investigating One Dimensional Motion page 2 of 2 Instantaneous Velocity and the Derivative Another way to determine the instantaneous velocity of the ball tossed in the air is to take the derivative of the position function In this example the equation for the ball s path is x 5t 2 20t The derivative of the position function or equivalently the instantaneous velocity of the ball is dx v dt d 5t 2 20t dt 2 1 1 1 2 5t 1 20t 10t 20 Evaluating the instantaneous velocity at t 0 results in v 10 0 20 20 indicating that the velocity is large and the ball is moving in the positive direction At t 2 the instantaneous velocity is zero v 10 2 20 20 20 0 At this point in time the ball is not moving For values of t 2 the instantaneous velocity is negative indicating that the ball is moving in the negative direction www thinkwell com info thinkwell com Copyright 2001 Thinkwell Corp All Rights Reserved 1774 rev 03 22 2001
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