Applications of the Equations of Motion - Book Report/Review Example

Introduction Equations of motion are equations that describe the motion of a particular object in terms of time. Equations of motion can be used to find out the displacement, velocity, time and acceleration of an object in motion. However these equations are useful only in the case of uniform acceleration and do not apply in the cases of non-uniform acceleration of objects (Frankel, 2011). These equations are also called as the Newton’s equations of motion. Contents Introduction 1 Contents 2 Literature Review 3 Derivation of the equations of motion 4 Experiment 6 Objective of the Experiment 6 Methodology 6 Result of the Experiment 7 Analysis of the Experiment 7 Graphical Data 10 Discussion 12 Conclusion 13 References 15 Literature Review The three equations of motions are as follows – V = u + at V^2 = u ^2 + 2as S = ut + (1/2) (a) (t^2) Where u = initial velocity of the object v= final velocity of the object a = acceleration of the object t = time taken to cover a distance s = displacement of the object The SI units used are as follows – u & v = m/s (metre/second) a = m/s^2(metre/second square) t= s (seconds) s = m (metres) Derivation of the equations of motion Let us suppose that any object that we are considering experiences an unbalanced force which causes the object to accelerate at a m/s^2.The force acts for a time period of t and the initial and final velocity of the object are u m/s and v m/s(Burton Linker , 1924). Now, we know that a = (v-u)/t multiplying both sided with t we get v –u = at Rearranging the above equation gives – V = u + at ------ 1 This equation is useful for calculating the velocity of a body which has experienced acceleration for a fixed period of time. This is our first equation of motion Now we know that V = s/t Thus distance travelled by the body is – S = vt Now as in our case the object is moving with uniform acceleration, we have to take into account the average velocity to find the distance covered Average velocity = (v+u)/2 Thus distance travelled is, s = average velocity * time s = [(u +v)/2]t Now from first equation of motion we have v = u + at ; we substitute this value in place of v and thus get the following result – s = [(2u + at)/2]t Rearranging the above equation , we get S = ut + (1/2)at^2...................2 This equation of motion can be used to calculate the distance covered by a body moving with uniform acceleration(Pearson , 2010). Now in order to derive the third equation we take equation 1 and square it , V^2 = (u +at )^2 V^2 = u ^2 + 2a(ut + (1/2)a*t^2) Now by using equation 2 ; we get V^2 = u^2 + 2as...............3 This equation of motion gives a relation between the final velocity of the object and the distance travelled by the object Experiment Objective of the Experiment The objective of this experiment is to demonstrate the use of the Newton’s equations of motion. We will use the equations of motion to calculate the unknown variables in the experiment and also verify the data with the help of these equations. Methodology A remote controlled car is being used for the purpose of the experiment. The start point and the end point for the car to move have been pre-defined. The distance between these points is the displacement of the car. For the purpose of the experiment this distance has been kept at 50m. The initial velocity of the car has been taken as 0 as the car was moved from rest. Once acceleration is provided the remote controlled car; the time taken by the car to reach the finish point is recorded with the help of a stop watch. The stopwatch was started as soon as the car crossed the start line and stopped when it reached the finish line. For the purpose of the experiment the force caused in the opposite direction by friction has been ignored. The experiment was conducted on a smooth marble surface to reduce the affect of friction as much as possible. In the end using the remote control of the car we also alter the acceleration of the car and try to show the failure of the equations of motion in the case of non-uniform acceleration. Result of the Experiment The following results were observed after the experiment – Distance travelled by the car 50 m Initial velocity of the car 0 m/s(car was at rest) Time taken to cover the distance 10 s In the second part of the experiment, the acceleration of the car was varied and the following results were obtained – Distance travelled by the car 50 m Initial velocity of the car 0m/s Time taken to travel the distance 13 s Analysis of the Experiment Now we will analyse the experiment conducted by us and see how equations of motion can be used to calculate the missing variables as well as verify these calculations Variables which are known to us are as follows – s(distance travelled) = 50 m t(time taken) = 10s u(initial velocity of the car) = 0m/s(starting from rest) We have these three variables which are known , we will use the equations of motion to calculate V(final velocity of the car) and also a(acceleration of the car). Now according to the second equation of motion , s = ut + (1/2)a*t^2 Now putting the known values of s and t in this equation we get , a = 1m/s^2 Thus the second equation of motion has been used to calculate the acceleration of the car Now using the first equation of motion , V = u + a*t We get V = 10m/s This is the final velocity of the car at the finishing line of the experiment. In order to ensure that the equations of motion stand and also to verify our calculation we will use the third equation of motion which is as follows – V^2 = u^2 + 2as Putting the values as V = 10 m/s , u = 0m/s and s = 10 m we should get the same result on both sides, Here we get , 100 = 100 , Thus the calculations performed as well as the experiment conducted by us has been successful . Now we will analyse the data of the second part of the experiment in which the acceleration of the car was varied , we will analyse if the equations are true when the acceleration is not constant. We will try to use the second equation of motion to find the acceleration just like in the first case , Putting s = 50m , u = 0m/s and t = 13s We get a = 0.59 m/s^2. However this acceleration we have calculated cannot be true as the acceleration was not constant. This can at best be said to be the average acceleration experienced by the car during the time it was in motion(Tmh , 2006). Now using the first equation of motion we will calculate the speed of the car at the finish line if we take a = 0.59 m/s^2.In this case ; V = 7.67 m/s Now we will try to calculate the distance using the third equation of motion. The distance calculated through the third equation should come out to be 50 m as we had taken 50 m as the distance of the movement of the car and the distance is known to us. Using the third equation of motion , V^2 = u^2 + 2 as Here , v = 7.67 m/s ; u = 0 m/s and a = 0.59 m/s^2 Using these values ; s = 49.85 m Thus the distance comes out to be 49.85 ;although the actual distance was 50 m. There is an error of 0.15 m which comes if we apply the equations of motion in the case of non-uniform acceleration. Although the error is small but we can say safely that the equations of motion are not valid in the case of non-uniform acceleration. Graphical Data The graph presented below shows the velocity of the car at different points of time of the experiment ; The experiment starts from the start line and ends at the finish line. The velocity at these two points is 0m/s and 10m/s respectively. The distance covered by the car can be calculated by finding the area of the graph between these two points. In order to calculate the distance covered we have to find the area of the triangle the formulae for which is , 1/2 * Base* height Now in this formulae, Base = time taken for the experiment which is 10 s Height = maximum speed reached by the car during the experiment .We have calculated this speed to be 10m/s according to the Newton’s equations of motions. Thus distance = ½ * 10 * 10 = 50m The distance traversed is thus 50m .This distance was known beforehand and thus verifies the experiment conducted by us. We can also use this graph to calculate the acceleration of the car during the experiment. We know that , Acceleration = change in velocity/Time The slope of the velocity line in the graph will thus give the acceleration of the object. In order to calculate this slope , we put the values which are known to us , Acceleration = (10 – 0)/10 = 1m/s^2 This was the same result which we had obtained through the equations of motion. Discussion The equations of motions can be used to predict information about the motion of an object if some of the information is known beforehand. This has been proved through the experiment of a remote controlled car that we have shown above. The Kinematic equations as the equations of motion are called help us to co-relate the different factors associated with motion – the initial speed of the object , the final speed of the object , the acceleration of the object , the time taken to cover the distance and finally the distance covered by the object. However the important thing which needs to be taken care in the use of these equations is that they are valid for uniform acceleration. In situations where the acceleration is not uniform; the equations of motions have limited use. The other factor which has not been considered during this experiment is the role of friction. We assumed that friction did not exist for the purpose of the experiment. This is unlikely to be the situation in real world .In real world problem; not only will friction play a role but the acceleration of the object will also be non-uniform. Another constraint of the equations of motion is that the motion should be only in a straight line. These equations are not valid for three dimensional motions of objects. However these equations of motions cannot be dismissed due to the constraints mentioned above. Although the equations will perfectly in an ideal condition but in non-ideal real world situation also these equations come close to the reality. They help us give a picture of the real situation is like. There are many applications of these equations. The most common application which we see daily is the speed gun used by traffic police to catch people who are over-speeding. The speed gun calculated the speed by taking pictures of the vehicle at two different frames of time. The time difference between these two shots allows the traffic police to calculate the speed by using the equations of motion. Equations of motions are in fact the backbone of physics with regards to speed and time calculations. Conclusion The equations of motion can be used to calculate various kinematic variables such as speed , time and the distance travelled. However the limitations with these equations of motion are that the acceleration needs to be constant and the line of motion should be straight. On the basis of the experiment conducted with the help of a remote controlled car we have proved the validity of these equations in the case of uniform acceleration and have also proved that equations throw up an error when we try to use them for non-uniform acceleration cases. References Burton Linker, J. (1924) Equations of motion..., London: Johns Hopkins University, p.24 - 45. Frankel, T. (2011) The Geometry of Physics: An Introduction, 3rd ed. London: Cambridge University Press, p.255 - 300. Pearson (2010) Physics Class 10,10th ed. Mumbai: Pearson Education India, p.25 - 93. Tmh (2006) Numerical Problems In Physics For Class Xi,2nd ed. New Delhi: Tata McGraw-Hill Education, p.48 - 100. Read More