{"id":3468,"date":"2021-07-23T17:01:26","date_gmt":"2021-07-23T11:31:26","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=3468"},"modified":"2024-09-30T08:10:44","modified_gmt":"2024-09-30T02:40:44","slug":"relative-motion-relative-velocity-in-one-two-dimensions","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/relative-motion-relative-velocity-in-one-two-dimensions\/","title":{"rendered":"Relative Motion | Relative Velocity in One & Two Dimensions"},"content":{"rendered":"\t\t
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Relative motion typically refers to, either the, relative position of one object as measured from the other object, or, relative velocity of one object with respect to another.<\/p>

So, we will start our discussion with relative position and you will find two practice problems on relative position to help improve your understanding<\/p>

Afterwards we will discuss relative velocity of one object with respect to another in one dimension followed by relative velocity in two dimensions where we will review rain man problem, the swimmer in river problem, and projectile problem where in projectile is projected from a moving platform at some relative velocity and projection angle and we will need to determine the initial velocity and projection angle with respect to the ground. Note that in all these 3 types of problems, we are either interested in establishing relative velocity vector in 2 dimensions or the velocity vector with respect to the ground.<\/p>

Afterwards we will review the ways to a) determine minimum distance between two objects moving in two dimensions and b) determine time to collision for two objects moving in two dimensions.<\/p>

So starting with relative position,<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Relative Position<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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For two particles, $A$ and $B$, moving along one dimension, say in the vertical direction, relative position of $A$ with respect to $B$ at some time $t$ would simply be $y_A(t) – y_B(t)$ and if we were to write the relative position of $B$ with respect to $A$ it would be $y_B(t) – y_A(t)$<\/p>

Now, let’s define the relative position of $A$ with respect to $B$ (or vice versa) in two or three dimensions.<\/p>

So, if the position vectors of $A$ and $B$ at some time $t$ are $\\overrightarrow{r}_A(t)$ and $\\overrightarrow{r}_B(t)$, then relative position vector of $A$ with respect to $B$ would be $\\overrightarrow{r}_A(t) – \\overrightarrow{r}_B(t)$ which we can write in the unit vector notation as $(x_A – x_B) \\hat{i} +$ $(y_A – y_B) \\hat{i} +$ $(z_A – z_B) \\hat{k}$<\/p>

It’s that simple and straightforward. Now let’s solve two problems that will challenge our problem solving skills. (Feel free to refer to solution videos to improve your understanding)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Q1. Two stones are thrown up simultaneously from the edge of a cliff $240 \\ m$ high with initial speed of $10 \\ m\/s$ and $40 \\ m\/s$ respectively. Plot the graph representing the time variation of relative position of the second stone with respect to the first? (Assume stones do not rebound after hitting the ground & neglect air resistance, $g = 10 \\ m\/s^2$)\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/i><\/span>\n\t\t\t\t\t\t\t\t<\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\tAnswer<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t

Correct graph is\"\"<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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