Fan and Motion Carts: A Study of Motion
Daniel K. Marble
Department of Mathematics, Physics, and Engineering, Box T-930,
TarletonStateUniversity, Stephenville, Texas76402,
Abstract:The position of a Pasco fan cart and Pasco motion cart have been measured as a function of time using a Pasco motion sensor. The position vs time graph of the motion cart shows that the cart moves at constant velocity whose magnitude depends on the setting of the speed controller while the position vs time graph for the fan cart shows that cart accelerates at rate dependent on the fan setting and the angle that it makes to the fan.
Theory: The position of an object is defined by a vector from the origin of a coordinate axis to the location of the object. The position vector for an object can be determined using the location of a sonic range finder as the origin of the coordinate axis and placing the x-axis in the direction of motion.The sonic ranger emits a short ultrasonic pulse at time to that travels to the object over a time t. Upon striking the object, the sonic pulse is reflected in a manner similar to a tennis ball striking a wall. The sound pulse then travels back to the sonic range finder taking the same amount of time t. The location of the object at time to+t is given from the definition of average velocity:
.(1)
For the sound pulse traveling, we haveand xo = 0 since the sonic ranger is at the origin. Thus the location of the cart is given by
.(2)
Thus, a sonic ranger actually measures time and then computes the position of the cart using the speed of sound and equation (2). As the speed of sound depends on temperature, the sonic range finder must first be calibrated by placing an object a known distance from the sonic range finder and using its time measurement to determine the speed of sound using equation (1).
A position vs time graph of an object is a powerful tool in analyzing motion. Over any time segment, the velocity of the object is constant if the graph is a straight line with the object’s velocity given by the slope of the graph. If the graph is not constant then the object is accelerating. The slope of the tangent line at any point on the graph gives the instantaneous velocity of the object at that point in time. By drawing tangent lines at various times, we can obtain a velocity vs time graph. This graph can then be used to determine the value of the acceleration of the object as a function of time.1
Experimental Procedure:A Pasco motion sensor was attached to one end of a 2.00 m Pasco dynatrack. A Pasco cart stop was placed on the other end of the dynatrack. The position as a function of time of both fan carts and motion carts was measured using the motion sensor and a Pasco 750 computer interface operating under Pasco Data Studio computer software. The speed of sound was first determined by placing a Pasco motion cart on the dynatrack at a distance of 1.00 m from the motion sensor and using the calibration command for the motion sensor in data Studio. The position vs time graphs for the motion cart and fan cart was then determined by placing the cart 25cm from the motion sensor and releasing the cart under a variety of operating conditions.
Results and Analysis: The position vs time graph for a Pasco motion cart is shown in Figure 1 for various settings of the variable speed control knob. In each case, the graph show a period of time in which the graph is a straight line with a non-zero slope. This indicates that the cart was traveling at a constant velocity during this time interval. The values for the slope give the velocity of the cart and are presented in Table 1. These results confirm that increasing the variable speed control knob on the cart does increase the slope of the position-time graph as expected and that the motion cart is a constant velocity cart. The right-hand side of each curve in Figure 1 has a slope of zero. This corresponds to the fact that the cart has hit the cart stop and is no longer moving in this time interval. The position of the cart at this time doesn’t correspond to the position of the cart stop because the sonic pulse is not reflected from the front of the cart. The difference between the final position of the cart and the position of the cart stop shows that the sonic pulse was reflected at a distance of 1.53 from the front of the cart.
Run / Speed (cm/s)Low / 6.320+/-0.003
Medium / 11.200+/-0.004
High / 25.800+/- 0.011
Figure 1Table 1
The position vs time graph for a Pasco fan cart is shown in Figure 2 for the fan setting on high with various settings of the fan angle. In all cases, the fan cart was accelerating since its graph is not a straight line during the time that the cart was moving. The velocity vs time graph for the fan cart for various fan angles is shown in Figure 3. The straight line portion of these curves demonstrates that the fan cart undergoes constant acceleration with a value that increases as the angle increases from 120 to 180 degrees.
Figure 2
Conclusion: The position vs time graph of a motion cart and a fan cart has been obtained for a variety of operating conditions. Our results indicate that a motion cart operates as a constant velocity cart while a fan cart operates in a constant acceleration condition.