Calculation of the Projectile Motion of a Ping Pong Ball with Drag Included - Lab Report Example

Introduction With increased complexity in mechanical systems it presents a lot difficulty or completely impossible attempting modeling the system by Newtonian mechanics and because of this the alternative to the classical method is resorting to Lagrangian mechanics.in this method there is introduction of Lagrangian unit expressed as L=T-V in which case T gives kinetic energy (KE) while V is potential energy. Using a Eula-Lagrange equation that is derived by minimizing action presents an easy way of having a solution of motion system equations. With the equation depending on the degree of freedom it exhibits the general coordinate system that is given by Calculation of the projectile motion of a ping pong ball with drag included The range of the ball is expressed as Calculating the movement catapult arm with a fixed mass (non-pivoting) This involves the application of the see saw model principle in which case both the projectile mass and the counter mass are to be attached to the beam. In physical terms it may obviously be seen that with the counter weight being much heavy when compared to the mass of the projectile, the former will take charge of the motion. Viewing the system from the long term angle it will; be realized that the system will have a periodic motion. The system is similar to a dual mass pendulum as similar to a metronome. As in this case the point of interest is in launching of a projectile motion in which case the periodic oscillation is not of interest. As the projectile motion of the equation upon release being in conformation with the simplified case of kinematics, then it is only important to put into consideration differential equation of projectile motion before release and this means application of Euler-Lagrangian equation will come into play. Position of the mass The first step is for the position of the two masses involved to be established and for the degree of freedom in x components and y components to be limited the positions as a function of  which is in turn a function of time. Putting into vectors the traced path of m1 the associated position vector p1 is given by Now the m2 coordinates are expressed by x2 = −l2 sin(_) and y2 = l2 cos(_). The position vector associated with m2, P2, path is given by : The Kinetic Energy (KE) and Potential Energy (PE) of system KE for a mass = In order to find velocity to be used in KE equation the position function is used by finding the derivative. To find m1 velocity we take the derivative for each component of P1. For m1, this gives us a velocity ( ) given by ( ) =< l1 cos(), l1  sin() > For velocity of m2 we have ( ) =< l2 cos(), l2  sin() > Now equipped with both KE and PE it is simple to calculate Lagrangian relationship Equations of motion Applying Euler-Lagrange equation to the Lagrangian relation yields the equation Solving this equation gives equation of motion as The parameters used in calculation were as follow Range Calculating the range will involve using of the x and y velocity of the projectile upon release od the ball For a case where air friction is not put into picture , a projectile will be expected to the following In these equations y0, y˙0, x0, and x˙0 are the initial conditions that should be found from the system at time of release and are functions of of  and . Through picking appropriate release times y can be set at 0 which is the moment the time of the ball hitting the ground and this helps in finding the solution to the kinematic equation. And through use of value of t that obtained in x equation, it will be possible to obtain the range of projectiles for different angles of ball release. The variation of range with angle of release is as shown in figure 1. From the figure it can be seen that the angle with maximum range is 38.5 with a range of 66.9m Ping pong with Hinged Counterweight This system resembles a double pendulum. The uniqueness of this system is that more of the PE in the counterweight is converted to KE of the projectile. In this system there is dependence on both and  meaning that the two motion equations are to be applied to get the solution. Mass position This involves the changing of only the counter mass position. Here it is like geometric addition of the initial position vector in the case of m1 with a new position vector being obtained from the new point. Thus we have Equations of motion With application of Euler-Lagrange equation to the Lagrangian we have the equation Solving this equation gives equation 1 of motion as Solution of ELE with respect to This gives as equation The following are the parameters used in calculation Figure 2 shows the variation of range with angle The figure reveals that the angle having a maximum range is 19 with the range being 394.5m which is about 6 times for the range involving seesaw model. In the excel result this is seen to be obeyed for the sling setup where we have at angle 22.5 the range being 2.17m compared to the values of 1.6 and 1nd 1.83 for 90 and 45 degree release angle in the same sling length set up. Location of pivot Choosing to use a mass ration of 100:1which is just picked arbitrary with a counter weight of 200g and projectile of 2g being used. The result figure 3. In the excel result it can be seen that for very small L2 the range was smallest and with an increase in L2 the range was also increased but the highest value of L2 resulted in a reduced range. This is an indication that the largest L2 was beyond the limit that is seen in the graph. Mass relationship When different masses of counterweights are used for the projectile of 2g the result will be as shown in the figure 5 below. This figure is obtained by use of a perfect engine model. The expectation would be that we should have the range having linear relationship with respect to mass ratio. This is seen to be true for the case involving small increases in counter weights where this have large effect on the range. However this trend changes at a certain point where increase in mass of counter weight does not have a significant effect. In the excel result it is observed that this general trend is observed where it can be seen that for the 300g mass the range in higher when compared to that of the 133g mass. Reference Siano, D. B. (2001).Trebuchet Mechanics, Read More