Question

A 75 kg box is dropped from the top of a tower. The height of the tower is 35 m. Calculate (i) the initial
potential energy of the box, (ii) its potential energy 15 m above the ground, (iii) the maximum value of
its kinetic energy and (iv) its kinetic energy 20 m below the top of the tower. (g = 9.8 m/s2)
(g = 9.8 m/s2)​

Answer 1

Given :-

• A 75 kg box is dropped from the top of a tower of height 35 m.
• Acceleration due to gravity, g = 9.8 m/s²

To Find :-

• Initial potential energy.
• Potential energy 15 m above the ground.
• Maximum value of kinetic energy
• Kinetic energy 20 m below the top of the tower.

Solution :-

First, Let us find the Initial potential energy at the top of the tower,

⇒ PE = mgh

⇒ PE = 75 × 9.8 × 35

⇒ PE = 25,725 J

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Here, We have

• Height, h = 15 m above the ground.
• g = 9.8 m/s²
• mass = 75 kg

⇒ PE = mgh

⇒ PE = 75 × 9.8 × 15

⇒ PE = 11,025 J

____________________________

Maximum value of the kinetic energy will be at the very bottom of the tower ( touching the ground ) where all the potential energy will be converted into Kinetic energy (According to law of conservation of energy)

So,

⇒ Maximum Kinetic energy = PE at the top of tower

⇒ Maximum Kinetic energy = 25,725 J

____________________________

Here, we have to find the kinetic energy 20 m below the top of the tower.

Let's find the potential energy at this point, we have

• Height, h = 35 - 20 = 15 m
• g = 9.8 m/s²
• mass = 75 kg

So,

⇒ PE = mgh

⇒ PE = 75 × 9.8 × 15

⇒ PE = 11,025 J

But, The total kinetic energy should be 25,725 J including the kinetic energy, according to the law of conservation of energy,

⇒ Total energy = PE + KE

⇒ 25,725 = 11,025 + KE

⇒ KE = 25,725 - 11,025

⇒ KE = 14,700 J

Answer 2

Given :

• Mass of the box = 75 kg

• Height of the tower = 35 m

To find :

• 1. The initial energy of the box

• 2. Its potential energy 15 m above the ground

• 3 . The maximum value of its kinetic energy

• 4. Its kinetic energy 20 m below the top of the tower

So , now first find the velocity of the box at the height of 35 m and 35 - 20 = 15 m.

First at 35 m :

$\sf : \implies \: \: \: \: \: {\sqrt{2 \times 9.8 \times 35}} \\ \\ \sf : \implies \: \: \: \: \: \: {\sqrt{19.6\times 35}} \\ \\ \sf : \implies \: \: \: \: \: \: {\sqrt{686}} \\ \\ \sf : \implies \: \: \: \: \: \: 26.195$

Now , at 15 m :

$</p><p>\sf : \implies \: \: \: \: \: \:{\sqrt{2 \times 9.8 \times 15}} \\ \\ \sf : \implies \: \: \: \: \: \:{\sqrt{19.6 \times 15}} \\ \\ \sf : \implies \: \: \: \: \: \:{\sqrt{294}} \\ \\ \sf : \implies \: \: \: \: \: \: 17.146$

Number 1 :

$\boxed{ \sf \: \: \: \: \: \: PI = mgh \: \: \: \: \: \: }$

Substitute all Values :

$\sf : \implies \: \: \: \: \: \:{PI=75kg \times 9.8m/s \times 35m} \\ \\ </p><p></p><p>\sf : \implies \: \: \: \: \: \:{PI=735 \times 35} \\ \\ </p><p></p><p> \sf : \implies \: \: \: \: \: \:{PI=25725J}$

So , Initial potential energy is 25725 joules.

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Number 2 :

$\boxed{ \sf \: \: \: \: \: \: pe = mgh \: \: \: \: \: \: }$

Substitute all values :

$\sf : \implies \: \: \: \: \: \:{Pe=75kg \times 9.8m/s \times 15m} \\ \\ \sf : \implies \: \: \: \: \: </p><p>{Pe=735 \times 15m} \\ \\ </p><p></p><p>\sf : \implies \: \: \: \: \: Pe =11025$

So , the potential energy at 15 m above the ground is 11025 joules .

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Number 3 :

$\boxed{ \sf \: \: \: \: \: \: Ke = \frac{1}{2}m {v}^{2} \: \: \: \: \: \: }$

Substitute all values :

$\sf : \implies \: \: \: \: \: Ke=\frac{1}{2} \times 75kg \times 26.195 \\ \\ </p><p>\sf : \implies \: \: \: \: \: Ke=37.5kg \times 26.195 \: \\ \\ \sf : \implies \: \: \: \: \: Ke=37.5kg \times 686 \\ \\ </p><p></p><p>\sf : \implies \: \: \: \: \: Ke=25725j \: \\ </p><p></p><p>$

So , the maximum kinetic energy is 25725 joules .

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Number 4 :

$\boxed{ \sf \: \: \: \: \: \: Ke = \frac{1}{2}m {va}^{2} \: \: \: \: \: \: }$

Substitute all values :

$\sf : \implies \: \: \: \: \:Ke=\frac{1}{2} \times 75kg \times 17.146 \\ \\ </p><p></p><p>\sf : \implies \: \: \: \: \:Ke=37.5kg \times 17.146 \\ \\ </p><p></p><p>\sf : \implies \: \: \: \: \:Ke=11025J \: \\ \\ </p><p>$

So , Kinetic energy below the 20 m of the tower is 11025 joules .