To derive the formula for the maximum height of a projectile launched vertically under the influence of gravity, we can use the equations of motion.
Assumptions:
- The only force acting on the projectile after launch is gravity.
- Air resistance is negligible.
- The upward direction is positive.
Variables:
- \( u \): Initial velocity (m/s) at which the projectile is launched upwards.
- \( g \): Acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)) acting downwards.
- \( h \): Maximum height (m) reached by the projectile.
Derivation Steps:
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Using the equations of motion: We can use the first equation of motion: \[ v^2 = u^2 + 2a s \] where:
- \( v \) is the final velocity (0 m/s at the maximum height).
- \( u \) is the initial velocity.
- \( a \) is the acceleration (which is \(-g\) since it's acting downwards).
- \( s \) is the displacement (which will be the maximum height \( h \)).
Plugging in these values into the equation: \[ 0 = u^2 - 2gh \]
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Rearranging for maximum height \( h \): \[ 2gh = u^2 \] \[ h = \frac{u^2}{2g} \]
Conclusion:
The formula for the maximum height \( h \) reached by a projectile launched vertically with an initial velocity \( u \) is: \[ h = \frac{u^2}{2g} \]
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