A toy helicopter is flying in a straight line at a constant...

A toy helicopter is flying in a straight line at a constant speed of $4.5 \mathrm{m} / \mathrm{s}$. If a projectile is launched vertically with an initial speed of $v_{0}=28 \mathrm{m} / \mathrm{s}$, what horizontal distance $d$ should the helicopter be from the launch site $S$ if the projectile is to be traveling downward when it strikes the helicopter? Assume that the projectile travels only in the vertical direction.

Key Concepts

Projectile Motion

Projectile motion involves the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. This concept requires the separation of motions into horizontal and vertical components with the understanding that horizontal motion has constant velocity while vertical motion is uniformly accelerated due to gravity.

Kinematic Equations

Kinematic equations are used to relate displacement, velocity, acceleration, and time in problems with constant acceleration. They are essential for calculating the height reached by a projectile, the time it takes to ascend or descend, and determining the conditions under which an object's velocity changes direction.

Time to Peak Height

The time to peak height is the duration needed for a vertically launched projectile to reach its highest point, where its upward velocity becomes zero before it starts descending. This concept is important for understanding when the projectile transitions from ascending to descending, which is critical when determining the conditions of impact.

Relative Motion Analysis

Relative motion analysis involves examining the motions of two or more objects from a common frame of reference, particularly when the objects are moving along different paths or directions. In the given problem, this involves comparing the horizontal motion of the helicopter to the vertical motion of the projectile to ensure that the collision occurs when the projectile is descending.

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