In a conversion from military to peacetime use, a missile...

In a conversion from military to peacetime use, a missile with maximum horizontal range $180 \mathrm{km}$ is being adapted for studying Earth's upper atmosphere. What is the maximum altitude it can achieve if launched vertically?

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Key Concepts

Projectile Motion

Projectile motion describes the motion of an object thrown into the air under the influence of gravity, where the horizontal and vertical components of the motion are analyzed separately. This concept is fundamental for understanding how objects travel through the air when launched at different angles, and it leads to formulas that predict outcomes like maximum range or maximum height.

Energy Conservation in Vertical Motion

When an object is launched vertically, its initial kinetic energy is gradually converted into gravitational potential energy until it reaches its maximum height. This conversion, governed by the conservation of energy, provides a method to calculate the maximum altitude by equating the initial kinetic energy to the gravitational potential energy at the peak.

Relationship Between Initial Velocity and Displacement

The initial velocity of a projectile is directly linked to its displacement in both horizontal and vertical directions. In an idealized scenario, formulas connecting the initial speed with the horizontal range for an optimally launched projectile also allow one to determine the maximum altitude when the motion is purely vertical, by using kinematic equations that relate velocity, acceleration, and displacement.

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