A particle is released from rest from a tower of height 3h. The ratio of the intervals of time to cover three equal heights h isa)t1:t2:t3= 3:2:1b)t1:t2:t3= 1:(√2-1):(√3-2)c)t1:t2:t3= √3:√2:1d)t1:t2:t3= 1:(√2-1):(√3-√2)Correct answer is option 'D'. Can you explain this answer?

Question

A particle is released from rest from a tower of height 3h. The ratio of the intervals of time to cover three equal heights h isa)t1:t2:t3= 3:2:1b)t1:t2:t3= 1:(&radic;2-1):(&radic;3-2)c)t1:t2:t3= &radic;3:&radic;2:1d)t1:t2:t3= 1:(&radic;2-1):(&radic;3-&radic;2)Correct answer is option 'D'. Can you explain this answer?

Answer

Let t₁, t₂, t₃ be the timings for three successive equal heights h covered during the free fall of the particle. Then

Subtracting (i) from (ii), we get

From (i),(iv) and (v), we get
t₁:t₂:t₃= 1:(√2-1) : (√3-√2)

Answer

Let t₁, t₂, t₃ be the timings for three successive equal heights h covered during the free fall of the particle. Then

Subtracting (i) from (ii), we get

From (i),(iv) and (v), we get
t₁:t₂:t₃= 1:(√2-1) : (√3-√2)

Answer

To solve this problem, we can use the equations of motion to find the time it takes for the particle to fall from each height.

The equation for the distance fallen by an object in free fall is given by:
d = 1/2 * g * t^2

Where:
d = distance fallen
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time

For the first height (3h), the distance fallen is 3h. So we have:
3h = 1/2 * g * t1^2

For the second height (2h), the distance fallen is 2h. So we have:
2h = 1/2 * g * t2^2

For the third height (h), the distance fallen is h. So we have:
h = 1/2 * g * t3^2

Now we can solve these equations to find the ratios of the time intervals.

Divide the first equation by the second equation:
(3h)/(2h) = (1/2 * g * t1^2)/(1/2 * g * t2^2)
3/2 = t1^2/t2^2

Take the square root of both sides:
√(3/2) = t1/t2

Divide the second equation by the third equation:
(2h)/(h) = (1/2 * g * t2^2)/(1/2 * g * t3^2)
2/1 = t2^2/t3^2

Take the square root of both sides:
√2 = t2/t3

So the ratio of the time intervals to cover three equal heights h is:
t1 : t2 : t3 = √(3/2) : 1 : √2

Therefore, the correct answer is option A) t1 : t2 : t3 = 3 : 2 : 1.

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