Vertical motion under gravity (application) *

Transcription

1 OpenStax-CNX module: m Vertical motion under gravity (application) * Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License.0 Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation. 1 Representative problems and their solutions We discuss problems, which highlight certain aspects of the study leading to the accelerated motion under gravity. The questions are categorized in terms of the characterizing features of the subject matter : Motion plots Equal displacement Equal time Displacement in a particular second Twice in a position Collision in air Motion plots Example 1 Problem : A ball is dropped from a height of 80 m. If the ball looses half its speed after each strike with the horizontal oor, draw (i) speed time and (ii) velocity time plots for two strikes with the oor. Consider vertical downward direction as positive and g = m / s. Solution : In order to draw the plot, we need to know the values of speed and velocity against time. However, the ball moves under gravity with a constant acceleration m / s. As such, the speed and velocity between strikes are uniformly increasing or decreasing at constant rate. It means that we need to know end values, when the ball strikes the oor or when it reaches the maximum height. In the beginning when the ball is released, the initial speed and velocity both are equal to zero. Its velocity, at the time rst strike, is obtained from the equation of motion as : * Version 1.: Oct, 008 :45 pm

2 OpenStax-CNX module: m14550 v Corresponding speed is : = 0 + gh The time to reach the oor is : v = ( gh ) = ( x x 80 ) = 40 m / s v = 40 m / s t = v u a = 40 0 = 4 s According to question, the ball moves up with half the speed. Hence, its speed after rst strike is : v = 0 m / s The corresponding upward velocity after rst strike is : v = 0 m / s Speed time plot Figure 1: surface. The ball dropped from a height looses half its height on each strike with the horizontal

3 OpenStax-CNX module: m Velocity time plot Figure : surface. The ball dropped from a height looses half its height on each strike with the horizontal It reaches a maximum height, when its speed and velocity both are equal to zero. The time to reach maximum height is : t = v u a = = s After reaching the maximum height, the ball returns towards oor and hits it with the same speed with which it was projected up, Corresponding speed is : The time to reach the oor is : v = 0 m / s v = 0 m / s t = s Again, the ball moves up with half the speed with which ball strikes the oor. Hence, its speed after second strike is : v = m / s

4 OpenStax-CNX module: m The corresponding upward velocity after rst strike is : v = m / s It reaches a maximum height, when its speed and velocity both are equal to zero. The time to reach maximum height is : t = v u a = 0 + = 1 s Example Problem : A ball is dropped vertically from a height h above the ground. It hits the ground and bounces up vertically to a height h/3. Neglecting subsequent motion and air resistance, plot its velocity v with the height qualitatively. Solution : We can proceed to plot rst by xing the origin of coordinate system. Let this be the ground. Let us also assume that vertical upward direction is the positive y-direction of the coordinate system. The ball remains above ground during the motion. Hence, height (y) is always positive. Since we are required to plot velocity.vs. height (displacement), we can use the equation of motion that relates these two quantities : v = u + ay During the downward motion, u = 0, a = -g and displacement (y) is positive. Hence, v = gy This is a quadratic equation. As such the plot is a parabola as motion progresses i.e. as y decreases till it becomes equal to zero (see plot below y-axis). Similarly, during the upward motion, the ball has certain velocity so that it reaches 1/3 rd of the height. It means that ball has a velocity less than that with which it strikes the ground. However, this velocity of rebound is in the upwards direction i.e positive direction of the coordinate system. In the nutshell, we should start drawing upward motion with a smaller positive velocity. The equation of motion, now, is : v = gy Again, the nature of the plot is a parabola the relation being a quadratic equation. The plot progresses till displacement becomes equal to y/3 (see plot above y-axis). The two plots should look like as given here :

5 OpenStax-CNX module: m Velocity displacement plot Figure 3: The nature of plot is parabola. 3 Equal displacement Example 3 Problem : A particle is released from a height of 3h. Find the ratios of time taken to fall through equal heights h. Solution : Let t 1, t and t 3 be the time taken to fall through successive heights h. Then, the vertical linear distances traveled are : h = 1 gt 1 h = 1 g ( t 1 + t ) 3h = 1 g ( t 1 + t + t 3 ) t 1 : ( t 1 + t ) : ( t 1 + t + t 3 ) :: 1 : : 3 t 1 : t 1 + t : t 1 + t + t 3 :: 1 : : 3 In order to reduce this proportion into the one as required in the question, we carry out a general reduction of the similar type. In the end, we shall apply the result to the above proportion. Now, a general reduction of the proportion of this type is carried out as below. Let :

6 OpenStax-CNX module: m t 1 : t 1 + t : t 1 + t + t 3 :: a : b : c From the rst two ratios, t1 a t1 + t = b = t1 + t + t3 c t1 a Similarly, from the last two ratios, t1 + t t1 = b a = t b a t1 + t b Now, combining the reduced ratios, we have : In the nutshell, we conclude that if = t1 + t + t3 t1 t c b = t3 c b t1 a = t b a = t3 c b t 1 : t : t 3 :: a : b a : c b Then, t 1 : t 1 + t : t 1 + t + t 3 :: a : b : c t 1 : t : t 3 :: a : b a : c b Applying this result as obtained to the question in hand, We have : t 1 : t 1 + t : t 1 + t + t 3 :: 1 : : 3 t 1 : t : t 3 :: 1 : 1 : 3 4 Equal time Example 4 Problem : Balls are successively dropped one after another at an equal interval from a tower. At the instant, 9 th ball is released, the rst ball hits the ground. Which of the ball in series is at ¾ of the height of the tower? Solution : Let t be the equal time interval. The rst ball hits the ground when 9 th ball is dropped. It means that the rst ball has fallen for a total time (9-1)t = 8t. Let n th ball is at 3/4th of the height of tower. Then, For the rst ball, For the nth ball, h = 1 g x 8t h 3h 4 = h 4 = 1 g x ( 9 n ) t

7 OpenStax-CNX module: m Combining two equations, we have : h = g x ( 9 n ) t = 1 g x 8t = 3gt ( 9 t ) = 16 9 n = 4 n = 5 5 Displacement in a particular second Example 5 Problem : A ball is dropped vertically from a tower. If the vertical distance covered in the last second is equal to the distance covered in rst 3 seconds, then nd the height of the tower. Consider g = m / s. Solution : Let us consider that the ball covers the height y n in n th second. Then, the distance covered in the n th second is given as : y n = u + g x ( n 1 ) = 0 + On the other hand, the distance covered in rst 3 seconds is : According to question, x ( n 1 ) = n 5 y 3 = ut + 1 gt = x x 3 = 45 m y n = y 3 n 5 = 45 n = 5 s Therefre, the height of the tower is : y = 1 x x 5 = 15 m 6 Twice in a position Example 6 Problem : A ball, thrown vertically upward from the ground, crosses a point A in time t 1. If the ball continues to move up and then return to the ground in additional time t, then nd the height of A from the ground.. Solution : Let the upward direction be the positive reference direction. The displacement equation for vertical projection from the ground to the point A, is : h = ut gt 1 In this equation, the only variable that we do not know is initial velocity. In order to determine initial velocity, we consider the complete upward motion till the ball reaches the maximum height. We know that the ball takes half of the total time to reach maximum height. It means that time for upward motion till maximum height is :

8 OpenStax-CNX module: m t 1 = t1+t As we know the time of ight, nal velocity and acceleration, we can know initial velocity : 0 = u gt 1 u = gt 1 = ( t1+t ) g Substituting this value, the height of point A is : h = ut gt 1 h = ( t1+t ) g x t 1 1 gt 1 h = gt1 + gt 1t gt 1 h = gt1t Example 7 Problem : A ball, thrown vertically upward from the ground, crosses a point A at a height 80 meters from the ground. If the ball returns to the same position after 6 second, then nd the velocity of projection. (Consider g = m / s ). Solution : Since two time instants for the same position is given, it is indicative that we may use the displacement equation as it may turn out to be quadratic equation in time t. The displacement, y, is (considering upward direction as positive y-direction) : y = ut + 1 gt 80 = ut + 1 x t = ut 5t 5t ut + 80 = 0 We can express time t, using quadratic formulae, Similarly, t = ( u ) + { ( u ) 4 x 5 x 80 } x 5 = u+ ( u 1600 ) According to question, Squaring both sides we have, t 1 = u ( u 1600 ) ( u t t 1 = 1600 ) 5 ( u 6 = 1600 ) 5 ( u 1600 ) = 30 u 1600 = 900 u = 50 m / s

9 OpenStax-CNX module: m Collision in air Example 8 Problem : A ball A is dropped from a height, h, when another ball, B is thrown upward in the same vertical line from the ground. At the instant the balls collide in the air, the speed of A is twice that of B. Find the height at which the balls collide. Solution : Let the velocity of projection of B be u and let the collision occurs at a fraction a of the height h from the ground. We should pause and make mental note of this new approach to express an intermediate height in terms of fraction of total height (take the help of gure). Collision in the mid air Figure 4: The balls collide as they move in same vertical line. Adhering to the conventional approach, we could have denoted the height at which collision takes place. For example, we could have considered collision at a vertical displacement y from the ground. Then displacements of two balls would have been : y and h y Obviously, the second expression of displacement is polynomial of two terms because of minus sign involved. On the other hand, the displacements of two balls in terms of fraction, are : ha and (1-a)h The advantage of the second approach (using fraction) is that displacements are stated in terms of the product of two quantities. The variable h appearing in each of the terms cancel out if they appear on ether side of the equation and we ultimately get equation in one variable i.e. a. In the nutshell, the approach using fraction reduces variables in the resulting equation. This point will be highlighted at the appropriate point in the solution to appreciate why we should use fraction?

10 OpenStax-CNX module: m14550 Now proceeding with the question, the vertical displacement of B is ah and that of A is (1- a)h. We observe here that each of the balls takes the same time to cover respective displacements. Using equation of constant acceleration in one dimension (considering downward vertical direction positive), For ball A, For ball B, ( 1 a ) h = 1 gt ah = ut + 1 gt The value of t obtained from the equation of ball A is : t = Substituting in the equation of ball B, we have : { ( 1 a ) h g } ah = u { ( 1 a ) h g } + ( 1 a ) h u { ( 1 a ) h g } = h u = { gh ( 1 a ) } Now, according to the condition given in the question, v A = v B v A = 4v B Using equations of motion for constant acceleration, we have : g ( 1 a ) h = 4 ( u gah ) gh g ( 1 a ) h = 4 { ( 1 a ) gah } 1 ( 1 a ) = { ( 1 a ) 4a } Recall our discussion on the use of fraction. We can write down the corresponding steps for conventional method to express displacements side by side and nd that the conventional approach will lead us to an equation as : ( h y ) = { h ( h y ) 4y } Evidently, this is an equation in two variables and hence can not be solved. It is this precise limitation of conventional approach that we switched to fraction approach to get an equation in one variable, ( 1 a ) = { 1 ( 1 a ) 4a } Neglecting, a =0,

11 OpenStax-CNX module: m The height from the ground at which collision takes place is : y = ah = h 3