The experiment aims at determining the value of gravitational acceleration (g) using simulation of objects under free fall, vertical acceleration, and objects in a projectile motion.
Bodies fall under the influence of gravitational forces. The force mg acts on the body. However, when such objects are falling under the influence of gravity, such motions are referred as free fall. The assumption underlying free fall motions is that they do not encounter air resistance. The motion of free falling objects is characterized by a downward acceleration. The equations of motions are modified as shown below.
V (t) = -gt +V0
y (t) = -1/gt2 +V0 t +y0
The initial velocity (V0) is zero, y0 is the initial position, and t is the time.
Also, gravitational force mg acts on objects on a projectile motion. This force is always directed on the vertical motion without affecting the horizontal motion. When the projectile is launched at an angle α, the horizontal component of the velocity Vx= U Cos α, while the vertical component Vy=U Sin α-gt, where U is the initial velocity. The y coordinate at any instance of time is given by the equation y= U Sin α- 1/2gt2 +y0, while that of x=x0 + U Cos α.
Graph of position y against time
The trajectory of the free fall object is a polynomial of order 2. The acceleration of the object can be obtained by calculating at any point on the polynomial. The 2. The negative sign indicates the motion is in direction to decrease the value of y. therefore, the gravitational acceleration g=8.2.
Graph of Velocity Vy Against time (t)
The graph of velocity of the free fall object against time is a straight line. This indicates a uniform acceleration as a result of gravitational forces; since only gravitational forces acts on the object. The acceleration is obtained by calculating the slope of the graph. In this case, the gravity g is given by the slope of the trend line; therefore g=9.272m/s2.
Graph of acceleration ay Against time (t)
The graph of the acceleration against time (t) is a cluster of points. The mean value of the acceleration is 13.83m/s2.
Graph of position (y) Against time (t)
For motion accelerated in the y direction, the graph of position y against time t is a polynomial of the equation y = -5.027t2 + 10.19t - 3.762. The acceleration at any point is given by the second derivative of the equation; however, the only deceleration acting on the body as a result of gravitational pull g. therefore, g=10.054m/s2.
Graph of Velocity Vy against time
The graph of velocity indicates that the velocity of the object decrease along the motion in y axis. This is because of the gravitational pull. The gravitational pull can be obtained by calculating the slope of the graph. In this case, the gravitational pull g=10.1m/s2.
Graph of acceleration against time
The graph of acceleration against time is a cluster around the mean gravitational acceleration of 14.52 m/s2. However, the mean value is affected by the outliers in the graph.
The projection experiences a gravitational pull until its velocity reaches zero at the turning point. Then, accelerates as it falls. The gravitational acceleration can be calculated from the equation of the polynomial by evaluating its second derivative with respect to t. therefore, g= -8.73m/s2.
The graph of the horizontal position against time is a straight line. This indicates that the acceleration along the horizontal motion is zero. This is a clear indication that the gravitation force mg acting on the object has no horizontal component; therefore, ax=0 throughout the motion.
The graph of Velocity in the horizontal direction X is almost horizontal indicating a constant velocity for the horizontal motion. As inferred from the above graph of position against time, the velocity is constant. Experimental errors resulting to outlier points contribute to the small variations in this graph.
The graph of velocity in the y direction against time indicates a constant acceleration. This is due to gravity and its value from the graph is g=-8.723m/s2.
The acceleration graph against time cluster shows the acceleration cluster on both sides of the mean value of 9.0466m/s2.
The method of position against time is not appropriate as may provide results with high deviation from the actual value of 9.8m/s2. This is because the error associated with the measurements is too large. To reduce this error, the velocity method is adopted. This method yields better results as compared to other variables of measuring gravity.
However, a quick method of finding the average value of the measured quantity may be adopted, it is not good as outlier values may affect the results highly.
Moreover, comparing the method of free fall and projectile motion, free fall is preferred to projectile. This is because of the dynamics of projectile motion, where g can be calculated effectively when the initial direction of the projectile is determined. While, the dynamics of free fall is simple to give the required results accurately.
Conclusion: the mean value of g obtained is 10.36m/s2, this is relatively higher above the theoretical value of 9.8m/s2. Experimental and computational errors can attribute to this variation. However, using the free fall method and computing the gravity from the relation of velocity against time can yield accurate results compared to other methods.