A football player kicks a ball with a speed of 22.4 m / s at an angle of 49.0^∘ above the horiz

step 1 We kick a ball with initial velocity 22 .4 meters per second at an angle theta knot equals 49 .0

A football player kicks a ball with a speed of 22.4 m / s...

A football player kicks a ball with a speed of $22.4 \mathrm{~m} / \mathrm{s}$ at an angle of $49.0^{\circ}$ above the horizontal from a distance of $39.0 \mathrm{~m}$ from the goal- post. a) By how much does the ball clear or fall short of clearing the crossbar of the goalpost if that bar is $3.05 \mathrm{~m}$ high? b) What is the vertical velocity of the ball at the time it reaches the goalpost?

A football player kicks a ball with a speed of $22.4 \mathrm{~m} / \mathrm{s}$ at an angle of $49.0^{\circ}$ above the horizontal from a distance of $39.0 \mathrm{~m}$ from the goal-
post.
a) By how much does the ball clear or fall short of clearing the crossbar of the goalpost if that bar is $3.05 \mathrm{~m}$ high?
b) What is the vertical velocity of the ball at the time it reaches the goalpost?

Key Concepts

Vertical Velocity Analysis

The vertical velocity of a projectile at any moment is determined by accounting for the initial vertical velocity and the acceleration due to gravity. By using the appropriate kinematic equation, one can find how much the vertical component of the velocity has been reduced (or increased in the downward direction) over a given time interval.

Time of Flight Determination Using Horizontal Motion

Since the horizontal component of the projectile's motion is unaffected by gravity, the time to reach a certain horizontal distance can be found by dividing that distance by the horizontal velocity component. This calculated time is then used in the vertical motion equations to determine factors like height or vertical velocity at that instant.

Projectile Motion

Projectile motion describes the two-dimensional movement of an object under the influence of gravity only, assuming negligible air resistance. In this type of motion, the object follows a parabolic trajectory that results from the combination of a constant horizontal velocity and a vertically accelerated motion due to gravity.

Decomposition of Initial Velocity

Breaking the initial velocity into horizontal and vertical components is essential for analyzing projectile motion. The horizontal component, found using the cosine of the launch angle, remains constant, while the vertical component, determined by the sine of the launch angle, is influenced by gravitational acceleration throughout the motion.

Application of Kinematic Equations

Kinematic equations for constant acceleration are used to relate displacement, velocity, time, and acceleration in projectile problems. These equations are applied separately to the horizontal and vertical motions—using constant velocity for horizontal displacement and uniformly accelerated motion for vertical displacement—to solve for unknown parameters.

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