Question

the ratio of times taken by a freely falling body to cover fourth metre and seventh metre is​

Answer 1

Given:

A body is falling freely

To find:

The ratio of times taken by a freely falling body to cover fourth metre and seventh metre

Solution:

We know that the equation of motion for freely falling bodies is given by,

$\boxed{h = ut \:+\: \frac{1}{2}gt^2 }$

where

h = height from where the body is falling freely

u = initial velocity = 0

g = acceleration due to gravity

t = time taken

From the question, we can see that we have to find the ratio of the time taken by the freely falling body to cover 4th meter and 7th meter i.e., we will find the values of the time when h = 4 meter and h = 7 meter.

Let's assume,

"t₄" = denote the time taken by the body to cover h = 4 m

"t₇" = denote the time taken by the body to cover h = 7 m

Now,

(1). Substituting h = 4 in the eq. of motion

4 = (0×t) + $\frac{1}{2}$gt₄²

⇒ 4 = $\frac{1}{2}$gt₄²

⇒ 8 = 9.8 × t₄² ..... [∵ g = 9.8 m/s²]

⇒  t₄² = 0.816

⇒ t₄ = 0.90 s

and

(2). Substituting h = 7 in the eq. of motion

7 = (0×t) + $\frac{1}{2}$gt₇²

⇒ 7 = $\frac{1}{2}$gt₇²

⇒ 14 = 9.8 × t₇²

⇒  t₇² = 1.428

⇒ t₇ = 1.19 s

∴ The ratio of the time taken by the free falling body is,

= $\frac{t_4}{t_7}$

= $\frac{0.90}{1.19}$

= 0.75

Thus, the ratio of times taken by a freely falling body to cover fourth metre and seventh metre is​ 0.75.

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Answer 2

(2 - √3) /  (√7 - √6) is the Ratio of times taken by a freely falling body to cover fourth meter and seventh meter

Explanation:

Using S = ut + (1/2)at²

u = 0  freely falling

a = g

S = 0  + (1/2)gt²

=> t²  = 2S/g

=> t = √(2S/g)

Time taken to cover 4th meter

time taken to cover 4 m  - Time Taken to cover 3m

=  √2(4)/g  - √2(3)/g

=  √(2/g) (2 - √3)

Time taken to cover 7th meter

time taken to cover 7 m  - Time Taken to cover 6m

=  √2(7)/g  - √2(6)/g

=  √(2/g) (√7 - √6)

Ratio of times taken by a freely falling body to cover fourth metre and seventh metre is​  =    (2 - √3) /  (√7 - √6)

= (2 - √3)( (√7 + √6)

= 2√7 + 2√6 - 3√2  - √21

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