A football player kicks a ball with a speed of 22.4 m / s at an angle of 49^∘ above the horizon

step 1 So here in first part of this question we can use equation as y is equals to v y t minus half g

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 $49^{\circ}$ above the horizontal from a distance of $39 \mathrm{~m}$ from the goal line. 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 $49^{\circ}$ above the horizontal from a distance of $39 \mathrm{~m}$ from the goal line.
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

Decomposition of Velocity

Breaking the initial velocity into horizontal and vertical components is essential in projectile motion. This allows one to analyze the two independent motions separately: the horizontal component determines the range and time, while the vertical component affects the maximum height and displacement at any given time.

Vertical Velocity

The vertical component of the velocity changes over time due to gravity. Calculating vertical velocity at a specific moment involves accounting for the initial vertical velocity and the gravitational acceleration acting over the elapsed time.

Vertical Displacement

This refers to the change in height of the projectile relative to its starting point, computed using the vertical component of the motion. It is crucial for determining whether the projectile clears or falls short of a certain vertical target, such as a crossbar.

Projectile Motion

This concept describes the motion of an object that is launched into the air and influenced only by gravity (ignoring air resistance). It combines horizontal and vertical motions which occur simultaneously, with the horizontal motion moving at constant velocity and the vertical motion undergoing constant acceleration due to gravity.

Kinematic Equations

Kinematic equations are mathematical formulas that describe the motion of objects under uniform acceleration. In projectile motion, these equations are used to calculate positions, velocities, and times by relating displacement, initial velocity, acceleration due to gravity, and time.

Time of Flight Calculation

Determining the time at which an object reaches a particular horizontal distance is crucial when analyzing projectile motion. By using the horizontal component of the velocity, one can compute the time of flight to that location, which is then applied to the vertical motion equations.

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