A football is kicked at ground level with a speed of 18.0 m/s at an angle of 31° to the horizontal. How much later does it hit the ground?

Understand the Problem

The question asks for the time it takes for a football, kicked at a specific speed and angle, to hit the ground. This involves calculations related to projectile motion.

Answer

The time it takes for the football to hit the ground is approximately $1.10$ seconds.

Steps to Solve

Identify the variables The speed of the football ($v$) is 18.0 m/s, and the angle ($\theta$) is 31°. We need to find the time of flight ($t$) until it hits the ground.
Calculate the vertical component of the velocity The vertical component of the initial velocity ($v_{y0}$) can be calculated using: $$ v_{y0} = v \cdot \sin(\theta) $$ Substituting the values, we have: $$ v_{y0} = 18.0 \cdot \sin(31°) $$
Calculate the total time of flight The time of flight ($t$) for a projectile launched and landing at the same height can be determined using the formula: $$ t = \frac{2 \cdot v_{y0}}{g} $$ Where $g$ is the acceleration due to gravity (approximately 9.81 m/s²). Substitute $v_{y0}$ from the previous step into this equation.
Final calculation First, calculate $v_{y0}$, and then substitute this value into the time formula: [ t = \frac{2 \cdot (18.0 \cdot \sin(31°))}{9.81} ]

The time it takes for the football to hit the ground is approximately: $$ t \approx 1.10 \text{ seconds} $$

More Information

The time of flight in projectile motion is influenced by the vertical component of the initial velocity and the acceleration due to gravity. This example illustrates how angles and initial speeds impact the duration of flight.

Tips

Forgetting to convert degrees to radians when using certain calculators.
Using the entire speed instead of the vertical component when calculating time of flight.
Not accounting for the effect of gravity appropriately in the formula.

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