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Jeff is part of a trekking team. As he climbs a hill, he drops his water bottle, which has a mass of 0.25 kilograms. The bottle slides down the hill and is moving at a velocity of 14 meters/second the instant it hits the ground. The vertical height from where Jeff dropped the bottle is [tex]\square[/tex] meters. Ignore friction, and use [tex]KE = \frac{1}{2} mv^2[/tex], [tex]PE = m \times g \times h[/tex], and [tex]g = 9.8 \, m/s^2[/tex].



Answer :

GinnyAnswer
To find the vertical height from which Jeff dropped the bottle, let's follow these steps:

1. Given Data:
- Mass of the bottle, [tex]\( m = 0.25 \)[/tex] kilograms
- Velocity when it hits the ground, [tex]\( v = 14 \)[/tex] meters/second
- Acceleration due to gravity, [tex]\( g = 9.8 \)[/tex] meters/second[tex]\(^2\)[/tex]

2. Calculate Kinetic Energy (KE):
- The formula for kinetic energy is [tex]\( KE = \frac{1}{2} m v^2 \)[/tex]
- Substituting the given values:

[tex]\[ KE = \frac{1}{2} \times 0.25 \, \text{kg} \times (14 \, \text{m/s})^2 \][/tex]

[tex]\[ KE = \frac{1}{2} \times 0.25 \times 196 \][/tex]

[tex]\[ KE = \frac{1}{2} \times 49 \][/tex]

[tex]\[ KE = 24.5 \, \text{Joules} \][/tex]

3. Relation Between Kinetic Energy and Potential Energy:
- When the bottle is at the height [tex]\( h \)[/tex] and is about to be dropped, it has potential energy (PE) equal to the kinetic energy (KE) it has just before hitting the ground (assuming no energy losses).

- The formula for potential energy is [tex]\( PE = m \times g \times h \)[/tex]
- Since [tex]\( PE = KE \)[/tex], we have:

[tex]\[ m \times g \times h = KE \][/tex]

[tex]\[ 0.25 \times 9.8 \times h = 24.5 \][/tex]

4. Solve for Height (h):

[tex]\[ 2.45 h = 24.5 \][/tex]

[tex]\[ h = \frac{24.5}{2.45} \][/tex]

[tex]\[ h = 10 \, \text{meters} \][/tex]

So, the vertical height from which Jeff dropped the bottle is [tex]\( 10 \)[/tex] meters.

Hence, the correct answer to fill in the box is:

[tex]\[ \boxed{10} \][/tex]

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