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Motion in Straight Line

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Physics Part I

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Physics

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12
Two stones are thrown up simultaneously from the edge of a cliff 200 m high with initial speeds of 15 m s-1and 30 m s-1. Verify that the graph shown in Fig. 3.27 correctly represents the time variation of the relative position of the second stone with respect to the first. Neglect air resistance and assume that the stones do not rebound after hitting the ground. Take g = 10 m s-2. Give the equations for the linear and curved parts of the plot.


bold For bold space bold first bold space bold stone comma space

Initial space velocity comma space straight u subscript 1 equals space 15 space straight m divided by straight s
Acceleration comma space straight a space equals space – straight g space equals space – space 10 space straight m divided by straight s squared

Now comma space using space the space second space equation space of space motion comma space

straight x subscript 1 space equals space straight x subscript 0 space plus space straight u subscript 1 straight t space plus space left parenthesis 1 divided by 2 right parenthesis at squared

space straight x subscript 0 space equals space 200 space straight m space is space the space height space of space the space cliff. space

Therefore comma space

straight x subscript 1 space equals space 200 space plus space 15 straight t space minus space 5 straight t to the power of 2 space end exponent space space space space space space space... left parenthesis straight i right parenthesis space

When space stone space hits space the space ground comma space straight x subscript 1 space equals space 0

therefore space – space 5 straight t squared space plus space 15 straight t space plus space 200 space equals space 0
rightwards double arrow space space space space space space space straight t squared space – space 3 straight t space – space 40 space equals space 0
rightwards double arrow space space space straight t to the power of 2 space end exponent – space 8 straight t space plus space 5 straight t space – space 40 space equals space 0 space

rightwards double arrow space straight t space left parenthesis straight t space – space 8 right parenthesis space plus space 5 space left parenthesis straight t space – space 8 right parenthesis space equals space 0 space

straight i. straight e. comma space space space space space straight t space equals space 8 space straight s space or space straight t space equals space – space 5 space straight s space

straight t space cannot space be space negative space as space the space stone space is space
projected space at space straight a space certain space time. space

Therefore comma space straight t space equals space 8 space sec

bold italic F bold italic o bold italic r bold space bold italic s bold italic e bold italic c bold italic o bold italic n bold italic d bold space bold italic s bold italic t bold italic o bold italic n bold italic e comma space

Initial space velocity comma space straight u subscript 2 space equals space 30 space straight m divided by straight s space
Acceleration comma space straight a space equals space – space straight g space equals space – space 10 space straight m divided by straight s squared

Now comma space using space the space second space equation space of space motion comma space

straight x subscript 2 space end subscript equals space straight x subscript 0 space end subscript plus space straight u subscript 2 straight t space plus space left parenthesis 1 divided by 2 right parenthesis at squared space

space space space space equals space 200 space plus space 30 straight t space minus space 5 straight t squared space space space space space space space space space space space space space space space space space space space space... left parenthesis ii right parenthesis

When space stone space hits space the space ground comma space straight x subscript 2 space equals space 0

So comma space
space space space space space – space 5 straight t squared space plus space 30 space straight t space plus space 200 space equals space 0 space
rightwards double arrow space space straight t to the power of 2 space end exponent – space 6 straight t space – space 40 space equals space 0 space
rightwards double arrow space space straight t squared space – space 10 straight t space plus space 4 straight t space plus space 40 space equals space 0 space

rightwards double arrow space space space space straight t space left parenthesis straight t space – space 10 right parenthesis space plus space 4 space left parenthesis straight t space – space 10 right parenthesis space equals space 0 space

rightwards double arrow space space space space space space space straight t space left parenthesis straight t space – space 10 right parenthesis space left parenthesis straight t space plus space 4 right parenthesis space equals space 0

rightwards double arrow space space space space space straight t space equals space 10 space straight s space or space straight t space equals space – space 4 space straight s

straight t space cannot space be space negative space again. space

therefore space straight t space equals space 10 space sec space

On space subtracting space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space

straight x squared space minus space straight x to the power of 1 space equals space left parenthesis 200 space plus space 30 straight t space minus 5 straight t squared right parenthesis space minus space left parenthesis 200 space plus space 15 straight t space minus 5 straight t squared right parenthesis

straight x subscript 2 space minus space straight x subscript 1 space equals 15 straight t space space space space space space space space space space space space space space... space left parenthesis iii right parenthesis

Equation space left parenthesis iii right parenthesis space represents space the space linear space path space of space both space stones. space

Due space to space this space linear space relation space between space left parenthesis straight x subscript blank to the power of 2 space end exponent end subscript – space straight x subscript 1 right parenthesis space and space straight t comma
the space path space remains space straight a space straight space line space till space 8 space straight s. space

At space straight t equals space 8 space sec comma space stones space are space seperated space
by space straight a space maximum space distance.

left parenthesis straight x subscript 2 space end subscript – space straight x subscript 1 right parenthesis subscript max space end subscript equals space 15 cross times space 8 space equals space 120 space straight m

After space 8 space straight s comma space only space second space stone space is space in space motion space
whose space variation space with space time space is space given space by space the space
quadratic space equation colon

straight x to the power of 2 space end exponent – space straight x to the power of 1 space equals space 200 space plus space 30 straight t space – space 5 straight t squared

Therefore comma space the space equation space of space linear space and space
curved space path space is space given space by comma

straight x subscript 2 space end subscript – space straight x subscript 1 space end subscript equals space 15 straight t comma space represents space linear space path.

straight x subscript 2 space end subscript ­ – space straight x subscript 1 space end subscript equals space 200 space plus space 30 straight t space – space 5 straight t squared comma space represents space the space curved space path. 180 Views

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What is motion?

When an object changes it's position with time, with respect to it's surroundings, the object is said to be in motion. 
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What is the most essential thing to study the motion of a body?

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When an object does not change its position with time, with respect to it's surroundings. In such a case the object is said to be at rest. 
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