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When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity?
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When a ball is thrown up vertically with velocity vo it reaches a maxi...
When a ball is thrown vertically with an initial velocity \( v_0 \), it reaches a maximum height \( h \). To understand how to triple this height, we need to analyze the relationship between initial velocity and maximum height.
Understanding Maximum Height
- The maximum height \( h \) achieved by the ball can be derived from the kinematic equation:
\[ h = \frac{v_0^2}{2g} \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
Relationship Between Height and Velocity
- To find the initial velocity required to triple the maximum height, let’s denote the new maximum height as \( H = 3h \):
\[ 3h = \frac{v^2}{2g} \]
- Substituting \( h \) into the equation gives:
\[ 3 \left( \frac{v_0^2}{2g} \right) = \frac{v^2}{2g} \]
- This simplifies to:
\[ 3v_0^2 = v^2 \]
Calculating New Velocity
- Solving for \( v \) (the new initial velocity):
\[ v = \sqrt{3} v_0 \]
Conclusion
- To triple the maximum height of the ball, it should be thrown with an initial velocity of:
\[ v = \sqrt{3} v_0 \approx 1.732 v_0 \]
- This indicates that the new velocity must be approximately 1.732 times the original velocity \( v_0 \) to achieve three times the maximum height.
Understanding Maximum Height
- The maximum height \( h \) achieved by the ball can be derived from the kinematic equation:
\[ h = \frac{v_0^2}{2g} \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)).
Relationship Between Height and Velocity
- To find the initial velocity required to triple the maximum height, let’s denote the new maximum height as \( H = 3h \):
\[ 3h = \frac{v^2}{2g} \]
- Substituting \( h \) into the equation gives:
\[ 3 \left( \frac{v_0^2}{2g} \right) = \frac{v^2}{2g} \]
- This simplifies to:
\[ 3v_0^2 = v^2 \]
Calculating New Velocity
- Solving for \( v \) (the new initial velocity):
\[ v = \sqrt{3} v_0 \]
Conclusion
- To triple the maximum height of the ball, it should be thrown with an initial velocity of:
\[ v = \sqrt{3} v_0 \approx 1.732 v_0 \]
- This indicates that the new velocity must be approximately 1.732 times the original velocity \( v_0 \) to achieve three times the maximum height.
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When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity?
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When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity?.
When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for When a ball is thrown up vertically with velocity vo it reaches a maximum height of h. If one wishes to triple the maximum height then the ball should be thrown with velocity?.
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