Question**: A projectile is launched at an angle of 45 degrees with an initial speed of 20 m/s. What is the maximum height reached by the projectile? (Use \( g = 9.8 \, \text{m/s}^2 \))

Question: A projectile is launched at an angle of 45° with an initial speed of 20 m/s. What is the maximum height reached by the projectile? (Use ( g = 9.8 , \ ext{m/s}^2 ))

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Understanding Projectile Motion and Maximum Height

When a projectile is launched at an angle, its motion can be broken down into horizontal and vertical components. The maximum height is determined solely by the vertical motion, specifically the component of velocity perpendicular to the ground.

Given:- Launch angle ( \ heta = 45^\circ )- Initial speed ( v_0 = 20 , \ ext{m/s} )- Acceleration due to gravity ( g = 9.8 , \ ext{m/s}^2 )

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Step 1: Vertical Component of Initial Velocity

Only the vertical component contributes to reaching maximum height. It is calculated using:

[v_{y} = v_0 \sin \ heta]

Substituting ( \ heta = 45^\circ ) and ( \sin 45^\circ = rac{\sqrt{2}}{2} ):

[v_y = 20 \ imes rac{\sqrt{2}}{2} = 10\sqrt{2} pprox 14.14 , \ ext{m/s}]

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Step 2: Use Kinematic Equation to Find Maximum Height

At maximum height, the vertical velocity becomes zero (( v_y = 0 )). Using the velocity equation:

[v_y^2 = u_y^2 - 2gh_{\ ext{max}}]

Where:- ( v_y = 0 ) (at peak)- ( u_y = 10\sqrt{2} , \ ext{m/s} )- ( h_{\ ext{max}} ) is the maximum height

Rearranging:

[0 = (10\sqrt{2})^2 - 2 \cdot 9.8 \cdot h_{\ ext{max}}]

[0 = 200 - 19.6 \cdot h_{\ ext{max}}]

Solving for ( h_{\ ext{max}} ):

[h_{\ ext{max}} = rac{200}{19.6} pprox 10.2 , \ ext{meters}]

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Final Answer

The projectile reaches a maximum height of approximately 10.2 meters.

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Why This Matters

Understanding projectile motion is essential in fields like engineering, sports, and physics. Knowing the peak altitude helps in predicting trajectories, optimizing launches, and analyzing real-world objects from cannonballs to sports balls.

Using precise values and correct kinematic equations ensures accurate calculations—especially important in design and safety applications.

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Keywords: projectile motion, maximum height formula, projectile launched at 45 degrees, physics projectile, vertical velocity componential, Katherine discover-insight, kinematic equations, gravity 9.8 m/s², trajectory calculation, physics problem solution

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Summary: For a projectile launched at 45° with 20 m/s, maximum height is approximately 10.2 meters when calculated using vertical motion principles and ( g = 9.8 , \ ext{m/s}^2 ).