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Answer :
To find the maximum height a ball reaches when thrown straight up, we can use the principle of conservation of energy. Here’s the step-by-step process:
1. Understand the Initial Situation:
- The ball has a mass of 5 kg.
- It is thrown with an initial velocity of 67 m/s.
- We'll use the acceleration due to gravity, [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex].
2. Calculate the Initial Kinetic Energy:
- The kinetic energy (KE) of an object is given by the equation:
[tex]\[
\text{KE} = \frac{1}{2} \times \text{mass} \times (\text{velocity})^2
\][/tex]
- Substitute in the given values:
[tex]\[
\text{KE} = \frac{1}{2} \times 5 \, \text{kg} \times (67 \, \text{m/s})^2
\][/tex]
- This initial kinetic energy calculates to 11222.5 Joules.
3. Apply Conservation of Energy:
- At the maximum height, all the kinetic energy will be converted into potential energy (PE).
- The potential energy is given by:
[tex]\[
\text{PE} = \text{mass} \times g \times \text{height}
\][/tex]
- At maximum height, [tex]\(\text{PE} = \text{KE}_{\text{initial}}\)[/tex], because energy is conserved.
4. Solve for the Maximum Height:
- Set the initial kinetic energy equal to the potential energy at the maximum height:
[tex]\[
11222.5 = 5 \times 9.81 \times \text{height}
\][/tex]
- Rearrange to solve for height:
[tex]\[
\text{height} = \frac{11222.5}{5 \times 9.81}
\][/tex]
- Calculate the height, which results in approximately 228.8 meters.
So, the ball reaches a maximum height of about 228.8 meters.
1. Understand the Initial Situation:
- The ball has a mass of 5 kg.
- It is thrown with an initial velocity of 67 m/s.
- We'll use the acceleration due to gravity, [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex].
2. Calculate the Initial Kinetic Energy:
- The kinetic energy (KE) of an object is given by the equation:
[tex]\[
\text{KE} = \frac{1}{2} \times \text{mass} \times (\text{velocity})^2
\][/tex]
- Substitute in the given values:
[tex]\[
\text{KE} = \frac{1}{2} \times 5 \, \text{kg} \times (67 \, \text{m/s})^2
\][/tex]
- This initial kinetic energy calculates to 11222.5 Joules.
3. Apply Conservation of Energy:
- At the maximum height, all the kinetic energy will be converted into potential energy (PE).
- The potential energy is given by:
[tex]\[
\text{PE} = \text{mass} \times g \times \text{height}
\][/tex]
- At maximum height, [tex]\(\text{PE} = \text{KE}_{\text{initial}}\)[/tex], because energy is conserved.
4. Solve for the Maximum Height:
- Set the initial kinetic energy equal to the potential energy at the maximum height:
[tex]\[
11222.5 = 5 \times 9.81 \times \text{height}
\][/tex]
- Rearrange to solve for height:
[tex]\[
\text{height} = \frac{11222.5}{5 \times 9.81}
\][/tex]
- Calculate the height, which results in approximately 228.8 meters.
So, the ball reaches a maximum height of about 228.8 meters.
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