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Path Or Trajectory

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<h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong>Path or Trajectory</strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:#000000">This article is about the path, path difference, trajectory, and</span><span style="color:#0e101a"> </span><a href="https://doubtnut.com/question-answer-physics/trajectory-equation-9773957" style="color:blue; text-decoration:underline">trajectory equations</a><span style="color:#0e101a">. </span><span style="color:#000000">After studying the article, we will be able to differentiate between path and trajectory. First of all, we will look at the concepts of the path and later on the trajectory and after that, we will solve some equations and problems.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>What is a Path in Physics?</strong></strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:#000000">The path is the fundamental concept of physics. It is defined as the total distance covered by an object. The total distance travelled by an object from the starting point. And we also refer the path as the</span><span style="color:#0e101a"> </span><a href="https://doubtnut.com/question-answer-physics/what-is-wavelength--46938418" style="color:blue; text-decoration:underline">wavelength</a><span style="color:#0e101a">, </span><span style="color:#000000">which shows the behaviour of the wave.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Path difference</strong></strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:#000000">If two sources of waves start from one point, they may reach a distinct location, when they reach a distant location depending on slight differences in the distance travelled they may be in phase or out phase. This&nbsp;<em>path difference</em>&nbsp;gives rise too many interesting phenomena like in the case of light interference patterns occurs and in the case of sound dead spots in auditoriums occurs. Path differences caused the colours on a soap bubble and oil slick.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Trajectory Definition</strong></strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">The distance travelled by an object in space is called a trajectory. The trajectory is also known as the flight path. The position of the object is maybe projectile and satellite. The path of the planets and asteroids are examples of satellite motion. Some familiar examples of the trajectory path are the path of a ball or a rock that is thrown. The path of aeroplanes is also known as the trajectory path.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong>Understanding Trajectory </strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">The trajectory is the distance travelled by an object in the space. The path of aeroplanes and the movement of the planet and asteroids are examples of trajectory. The trajectory is also known as the flight path. A path followed by a moving object that passes through space is known as a trajectory. In mathematically trajectory is the position of the object in some particular time. A simple example is a ball or rock that was thrown upwards. The path taken by the stone is determined by the gravitational forces and resistance of air.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Some more common examples are a fired bullet from a gun, rotation of the earth, movement of asteroids, etc.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Trajectory equation</strong></strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">The trajectory equation is derived by using the following formula.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Y = Xtan&theta;&minus;gX2 / 2v2cos2&theta;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">here,&nbsp;<em>Y</em>&nbsp;is the horizontal components, X is the vertical component,&nbsp;<em>g&nbsp;</em>= value of gravitation,&nbsp;<em>v&nbsp;</em>= initial velocity, &theta; = angle of inclination of from horizontal axis.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">There are also equations related to the trajectory which are given as follow:</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Time of Flight = 2v0sin&theta; / g</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Where &lsquo;g&rsquo; is the value of the gravitational force.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Maximum height reached (H) = V2 sin2&theta; / 2g</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Where &lsquo;g&rsquo; is the value of gravitation.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Horizontal Range (R) =V2sin2&theta;/g</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Where, &lsquo;g&rsquo; is the value of gravitation, V = initial Velocity, sin &theta; = y-axis vertical component, cos &theta; = x-axis horizontal component.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Now for the better understanding, let us solve some examples on the trajectory.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Related Videos:</strong></span></span></span></p> <ul> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><a href="https://doubtnut.com/question-answer-physics/look-at-the-graphs-a-to-d-carefully-and-state-with-reasons-with-of-these-cannot-possibly-represent-o-20479035" style="color:blue; text-decoration:underline">Look at the graphs (a) to (d) carefully and state, with reasons, with of these cannot possibly represent one dimensional motion of a particle.</a></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><a href="https://doubtnut.com/question-answer-physics/suggest-a-suitable-physical-situation-for-each-of-the-following-graph-fig-20479040" style="color:blue; text-decoration:underline">Suggest a suitable physical situation for each of the following graph . Fig.</a></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><a href="https://doubtnut.com/question-answer-physics/the-given-acceleration-time-graph-represents-which-of-the-following-physical-situation-32543767" style="color:blue; text-decoration:underline">The given acceleration-time graph represents which of the following physical situation?</a></span></span></li> </ul> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Solved Examples</strong></strong></span></span></span></h2> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Example 1:</strong> John throws a rock at an angle of 60 degrees. If it moves at the rate of 4 m/s and jack catches it after 5 seconds. Calculate the distance covered by it.</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Solution:</strong></span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Given data:</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">&nbsp;&theta;=60 degree</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Velocity = 4 m/s</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">&nbsp;Time = 5 seconds</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">The horizontal distance is found by:</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">X =X V<sub>0</sub> t</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">X = 4 m/s &times; 5 s</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">&nbsp;X = 20 m</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = (20) Tan 600 &ndash; [(9.8) (20)2 / (2) (4)2 cos2 60]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = (20) (1.7320) &ndash; [(9.8) (20) (20) / (2) (16) (0.25)]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = 34.64 &ndash; [3920 / 8]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = 34.64 &ndash; 490</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = &ndash; 455.36 m</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Example 2:&nbsp;</strong></span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Jack throws a basketball at an angle of 60 degrees. If it moves at the rate of 5 m/s and john catches it after 3 seconds. Calculate the distance covered by it.</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Solution:</strong></span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Given, data:</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">&nbsp;&theta;=60 degree, velocity= 5 m/s, time = 4 s</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">The horizontal distance is found by:</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">X =X V0 t</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">X = 5 m/s &times; 4 s</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">&nbsp;X = 20 m</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = (20) Tan 600 &ndash; [(9.8) (20)2 / (2) (5)2 cos2 60]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = (20) (1.7320) &ndash; [(9.8) (20) (20) / (2) (25) (0.25)]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = 34.64 &ndash; [3920 / 12.5]</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Y = 34.64 &ndash; 313.6</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Y = &ndash; 278.96 m</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Time of Flight</strong></strong></span></span></span></h2> <p><span style="font-size:16px"><span style="color:#000000"><span style="font-family:Times New Roman,Times,serif">&nbsp;The Time of Flight is a method for measuring the distance between an observer and object, based on the time. The formula which is used to find the time of flight is given as follow:</span></span></span></p> <p><span style="font-size:16px"><span style="color:#000000"><span style="font-family:Times New Roman,Times,serif">Time of Flight = 2V<sub>0</sub>sin&theta; / g</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:#000000">Where &lsquo;g&rsquo; is the value of gravitation.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Maximum Height Formula</strong></strong></span></span></span></h2> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">In the action of gravity, a projectile is an object that is given initial velocity. And we can also say that the highest vertical position along its trajectory is called the maximum height of the object. The maximum height of the projectile depends on the&nbsp;<em>V (</em>initial velocity), the launch angle&nbsp;<em>&theta;</em>, and the acceleration due to gravity. The unit used for maximum height in meters (<em>m</em>).</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Maximum height = (initial velocity) 2 (sin &theta;2) / 2(acceleration due gravity)</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H= V2 sin2 &theta; / 2g</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Where H is the maximum height in meters</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><em>V0</em>&nbsp;is the initial velocity;<em>&nbsp;&lsquo;g&rsquo;</em>&nbsp;is the acceleration due to gravity, whose value is 9.80 m /s<sup>2</sup><em>,</em>&nbsp;<em>&theta;</em>&nbsp;is the angle of the initial velocity from the horizontal plane in radians or degrees.</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Solved Example&nbsp;</strong></span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">A firefighter aims a fire hose, toward a fire in a skyscraper. The water leaving the hose has a velocity of 30.0&nbsp;<em>m/s</em>. If the firefighter holds the hose at an angle of 70.5 degrees, find the maximum height of the water stream?</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Solution&nbsp;</strong></span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">The maximum height can be found using the formula:</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">Maximum height reached (H) = V2 sin2&theta; / 2g</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H = (30.0)2 sin2 (70.5) / 2(9.8)</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H = (30.0)2 (0.9426)2 (70.5) / 2(9.8)</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H = (900) (0.888) (70.5) / 19.6</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H= 56343.6 / 19.6</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">H = 2874.6 m</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif">The maximum height is 2874.6&nbsp;<em>m</em>.</span></span></span></p> <p><span style="color:#000000"><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><strong>Important Chapters:</strong></span></span></span></p> <ul> <li><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><span style="color:black"><a href="https://doubtnut.com/physics-ncert-solutions" style="color:blue; text-decoration:underline">NCERT Solution for Physics</a></span></span></span></li> <li><span style="font-size:16px"><span style="font-family:Times New Roman,Times,serif"><span style="color:black"><a href="https://doubtnut.com/physics-ncert-solutions/class-11-physics-chapter-3-motion-in-a-straight-line-exercise-1" style="color:blue; text-decoration:underline">NCERT Solution for Class 11 Physics Chapter Motion In A Straight Line</a></span></span></span></li> </ul>

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