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A body rolls down a staircase of \[5\]steps. Each step has height \[0.1m\]and width \[0.1m\]. With what velocity will the body reach the bottom.
A) \[\sqrt {\dfrac{5}{2}} m/s\]
B) \[\sqrt {\dfrac{1}{2}} m/s\]
C) \[2\sqrt 2 m/s\]
D) \[\dfrac{5}{{\sqrt 2 }}m/s\]

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Last updated date: 19th Jun 2024
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Answer
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Hint: Here in this question firstly we will calculate the total height and width and then we will apply equation of motion as \[d = \dfrac{{g{t^2}}}{2}\] and \[d = ut\] where \[d\], \[u\] and \[t\]are distance ,speed and time respectively.

Complete step by step answer:
As we know in this question we are given with us the height and width of each step as \[0.1m\] and \[0.1m\]. So to calculate total height and width multiply given heights and widths by \[5\].
So total height and width are
\[h = 5 \times 0.1\]
\[h = 0.5\], so now for width do the same as height
\[w = 5 \times 0.1\]
\[w = 0.5\]
So now to calculate time we have \[d = u \times t\],so after substituting values we get
\[0.5 = u \times t\]
\[t = \dfrac{{0.5}}{u}\],so now substitute the value of time in \[d = \dfrac{{g{t^2}}}{2}\]to calculate velocity with which it has fallen on ground, so we get
\[0.5 = \dfrac{{10 \times {{\left( {\dfrac{{0.5}}{u}} \right)}^2}}}{2}\]
\[0.5 \times 2 = \dfrac{{10 \times 0.25}}{{{u^2}}}\]
\[{u^2} = \dfrac{{2.5}}{1}\]
\[u = \sqrt {\dfrac{5}{2}} \]

So, The correct option is A.

Additional Information:
In such questions, an important thing that we consider is the assumption that firstly, that the horizontal velocity of the ball is constant throughout the process and air resistance or friction are not present. Secondly, we assumed the fact that the ball does not bounce back after every vertical collision with the steps, if we had not assumed the same, the question would also need to consider the time taken by the ball to rise to a specific height considering how much energy the ball loses after every collision. Proceed with the explained process if the question provides you with the needed information about energy loss after every collision with the steps.

Note: As in this question we have used initial velocity \[0\] and in newton’s equations of motion is has used acceleration of gravity as g and ignoring resistance by air that is we are assuming ideal stage that there is no force acting to body other than the force of gravity and ignoring the air resistance that can also move the ball in any other side of staircase.