A soccer player kicks a ball with an initial speed of 14 m / s at an angle θwith the horizontal

step 1 So in this problem, we're looking at an application of our trick. We have a soccer ball. We have

A soccer player kicks a ball with an initial speed of 14 m /...

A soccer player kicks a ball with an initial speed of 14 $\mathrm{m} / \mathrm{s}$ at an angle $\theta$ with the horizontal (see the accompanying figure). The ball lands $18 \mathrm{m}$ down the field. If air resistance is neglected, then the ball will have a parabolic trajectory and the horizontal range $R$ will be given by $$ R=\frac{v^{2}}{g} \sin 2 \theta $$ where $v$ is the initial speed of the ball and $g$ is the acceleration due to gravity. Using $g=9.8 \mathrm{m} / \mathrm{s}^{2},$ approximate two values of $\theta,$ to the nearest degree, at which the ball could have been kicked. Which angle results in the shorter time of flight? Why?

A soccer player kicks a ball with an initial speed of 14 $\mathrm{m} / \mathrm{s}$ at an angle $\theta$ with the horizontal (see the accompanying figure). The ball lands $18 \mathrm{m}$ down the field. If air resistance is neglected, then the ball will have a parabolic trajectory and the horizontal range $R$ will be given by
$$
R=\frac{v^{2}}{g} \sin 2 \theta
$$
where $v$ is the initial speed of the ball and $g$ is the acceleration due to gravity. Using $g=9.8 \mathrm{m} / \mathrm{s}^{2},$ approximate two values of $\theta,$ to the nearest degree, at which the ball could have been kicked. Which angle results in the shorter time of flight? Why?

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 only gravity (assuming negligible air resistance). This motion is characterized by a parabolic trajectory, where the horizontal and vertical motions can be analyzed independently using kinematics.

Horizontal Range Formula

The horizontal range in projectile motion is given by the formula R = (v²/g) sin 2?. This formula is derived from the kinematic equations and encapsulates how the initial speed, angle of projection, and gravitational acceleration together determine how far the projectile will travel horizontally before landing.

Complementary Angles in Projectile Motion

For any given initial speed and horizontal range, the angle of projection can have two solutions because sin 2? is the same for two complementary angles (? and 90°-?). This reflects the symmetry in the projectile’s path, where two different launch angles can yield the same horizontal range though with different flight characteristics.

Time of Flight

The time of flight of a projectile depends on the vertical component of its initial velocity. A higher launch angle increases the time the projectile spends in the air by increasing the vertical displacement, whereas a lower launch angle results in a faster landing. Therefore, among two complementary angles that yield the same range, the lower angle produces a shorter time of flight because it has a smaller vertical component.

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