After finding the acceleration and the time required for the pass, use Maple to plot the position of both the automobile and the truck as a function of time. From the data supplied in the figure, calculate the acceleration of the automobile during the pass and the time required for the pass. The diagram above shows the passing ability of an automobile at low speed. At the end you will find an animation from the truck's point of view that reveals the answer to the problem. Perhaps you can even solve it a third time, from the point of view of the car. You should solve it twice - once from the point of view of an external stationary observer, and again from the point of view of the truck. A good example of this is provided by the following "classic" problem. This can be especially useful when solving motion problems if one of the objects in the problem is moving with constant velocity relative to the "external" frame. It is a basic principle of physics - called Gallilean relativity - that the laws of physics are the same when measured with respect to two reference systems that are moving at constant velocity with respect to one another. Relative acceleration is defined the same way. Relative velocity is defined in the obvious way as the derivative of the relative position - which is of course the difference of the velocities of the objects. In the simple example in the preceding section, you have seen that the relative position of two objects moving in one dimension is simply the difference in their positions (measured in any reference system). How are the two related?Ī problem inspired by the driver's manual and the white Bronco You should also view the action from the point of view of the second object. Problem 3: In the worksheet, the motion is viewed from the first object. You can explore the motion of the two objects, both from an "external" point of view and from the point of view of one or the other of the objects in this Maple worksheet. Problem 2: When does the second object hit the ground? Problem 1: What function describes the motion of the second object for 0 5. The second is kept on the ground until time t = 5 seconds, and then it is projected upward with initial velocity 320 feet per second. The first is projected upward when time t = 0 with initial velocity 400 feet per second - so its height at time t is given by the function y( t) described above. If we decide to measure time in seconds and height in feet, and if the object is projected upward at time t = 0 with a velocity of 400 feet per second, then the function that gives the height of the object at time t isĪctually, this function only works for t between 0 and 25 seconds - because 25 seconds after the object is projected upward, it hits the ground. Its motion can therefore be described by one function, y( t), which gives the height of the object above the earth for any time t. Depending on whether the motion is taking place along a line (one dimension), in a plane (two dimensions) or in space (three dimensions), we use one, two or three functions to specify the position of the object at any time.Īn object that is projected straight up from the surface of the earth and then is subject only to gravity moves within a straight vertical line. Usually, the independent variable in these equations is t, for time. One-dimensional projectile motion and warmup exercisesĮquations are used to describe the motion of objects. This module concerns the process of changing the point (and direction) of reference from which the motion is viewed - this is important for solving physics problems, designing automatic pilots and other robotic devices, and video games. Depending on whether the motion is taking place along a line (one dimension), in a plane (two dimensions), or in space (three dimensions), we use one, two or three functions to specify the position of the object at any time. Usually the independent variable in these equations is t, for time. Larry Gladney is Associate Professor of Physics and Dennis DeTurck is Professor of Mathematics, both at the University of Pennsylvania.Įquations are used to describe the motion of objects. Maple is not required for the use of the ideas in this module, but it is required for opening and executing the downloadable files. Any other CAS can be used instead (e.g., Mathematica, Mathcad, etc.) as long as the user is familiar with that CAS system. Note: The activities in this module make reference to the computer algebra system (CAS) Maple, and links are provided to download Maple files.
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