Question

d. A stone thrown vertically upwards with initial velocity u reaches a height 'h' before coming down. Show that the time taken to go up is same as the time taken to come down.
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Answer 1

Given:-

• A stone thrown vertically upwards with initial velocity " u ".

• Reaches a height " h " before coming down.

To Prove:-

• The time taken to go up is same as the time taken to come down.

Proof:-

Here, We have to use the equation of motions,

v = u + at               -------------------- (1)

v² = u² + 2as         -------------------- (2)

Where v is the final velocity after time t, u is initial velocity, a is acceleration and s is distance travelled in time t.  

 

Let u be the initial velocity of stone.

when the stone reaches the maximum height after time t , its final velocity is zero.

When the stone travelling vertically upwards, it is experiencing the retardation due to gravity.

hence, we write the above equation of motion (1) as,  

$0 = u - g \times t \ \ or \ \ t = \dfrac{u}{g}$       -------------------- (3)

equation (2) will be written as,   0 = u2 - 2×g×h, or

 

$u = \sqrt{2 \times g \times h}$       ------------------- (4)

by substituting u from equation (4) in equation (3), we have t

$\dfrac{u}{g} = \dfrac{\sqrt{2 \times g \times h}}{g} = \sqrt{\dfrac{2 \times h}{g}}$     ------------------- (5)

After reaching maximum height, the stone descends with zero initial velocity,  accelerated downwards due to gravity and reaches the ground after time t'.

 

Equations (1) and (2) are written as follows for downward travel

$v = 0+gt'\ \ or\ \ t' = \dfrac{v}{g}$         -------------- (6)

$v^{2} = 0 + 2gh$

$v = \sqrt{2 \times g \times h}$             ------------------- (7)

from equation (4) and equation (7), we can conclude that initial velocity and  final velocity when the stone reaches the ground are equal.

 

by substituting for v from equation (7) in equation (6), we get

$t' = \dfrac{v}{g} = \dfrac{\sqrt{2 \times g \times h}}{g} = \sqrt{\dfrac{2 \times h}{g}}$       ----------------------- (8)

from equation (5) and equation (8),  time of ascending the height h is equal to time of descending from same height h.

Hence, The time taken to go up is same as the time taken to come down.

                    ------------------------- Proved ---------------------------