A badminton shuttlecock is struck by a racket at a 35^∘ angle, giving it an initial speed of 10

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A badminton shuttlecock is struck by a racket at a 35^∘...

A badminton shuttlecock is struck by a racket at a $35^{\circ}$ angle, giving it an initial speed of $10 \mathrm{~m} / \mathrm{s}$. How high will it go? How far will it travel horizontally before being contacted by the opponent's racket at the same height from which it was projected?

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and moves under the influence of gravity alone, after any initial propulsive force has been applied. This fundamental concept involves analyzing the motion in both vertical and horizontal directions independently, using kinematic equations and the assumption of constant acceleration due to gravity.

Vector Decomposition

Vector decomposition is the process of breaking the initial velocity into horizontal and vertical components using trigonometric functions like cosine and sine. This is crucial because, in projectile motion, the horizontal and vertical motions are independent: the horizontal motion typically has constant velocity while the vertical motion involves constant gravitational acceleration.

Maximum Height Calculation

The maximum height reached by a projectile is determined by the vertical component of its initial velocity and the acceleration due to gravity. By setting the final vertical velocity to zero at the apex, one can use kinematic equations to solve for the maximum height, which depends solely on the initial vertical velocity and gravitational acceleration.

Projectile Range

The range of a projectile is the horizontal distance it travels before returning to its initial vertical level. It is calculated by using the horizontal component of the velocity and the total time of flight. The range equation integrates both the initial speed and the angle of launch, showing how these affect the overall distance covered.

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