(II) An athlete executing a long jump leaves the ground at a 28.0^∘ angle and travels 7.80 m . (

step 1 So for problem 24, we are given the information that someone executes a long jump with an initia

(II) An athlete executing a long jump leaves the ground at a...

(II) An athlete executing a long jump leaves the ground at a $28.0^{\circ}$ angle and travels 7.80 $\mathrm{m}$ . (a) What was the takeoff speed? (b) If this speed were increased by just $5.0 \%,$ how much longer would the jump be?

Key Concepts

Projectile Motion

Projectile motion is the two-dimensional motion of an object under the influence of gravity, where the only acceleration acting on the object is due to gravity (ignoring air resistance). It consists of independent horizontal and vertical motions, with the horizontal component having constant velocity and the vertical component having constant acceleration due to gravity.

Range Equation

The range equation expresses the horizontal distance traveled by a projectile launched from the ground in terms of its initial speed, launch angle, and gravitational acceleration. The formula, R = (v?² sin(2?))/g, is used to connect the takeoff speed and angle to the total distance covered, making it crucial in analyzing projectile motion in sports like the long jump.

Vector Decomposition

Vector decomposition involves breaking down an object's initial velocity into horizontal and vertical components, using trigonometric functions. The horizontal component, v? cos?, determines the distance traveled over time, while the vertical component, v? sin?, influences the time the projectile spends in the air. This separation is essential for understanding and solving problems in two-dimensional kinematics.

Sensitivity of Range to Initial Speed

The range of a projectile is proportional to the square of its initial speed, meaning that small changes in speed can lead to significant changes in the range. This quadratic relationship is important when analyzing how variations in takeoff speed, such as a 5% increase, affect the overall distance traveled by the projectile.

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