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 is asking for the time it takes for a football, kicked at a specific speed and angle, to hit the ground after being kicked. We will use the principles of projectile motion to find the answer.
Answer
The football hits the ground after approximately $1.89 \, \text{s}$.
Answer for screen readers
The football hits the ground after approximately ( 1.89 , \text{s} ).
Steps to Solve
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Identify Variables Let ( v_0 = 18.0 , \text{m/s} ) be the initial speed and ( \theta = 31^\circ ) be the angle of projection.
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Calculate Time of Flight The time of flight ( T ) for a projectile can be calculated using the formula: $$ T = \frac{2 v_0 \sin(\theta)}{g} $$ Where ( g = 9.81 , \text{m/s}^2 ) is the acceleration due to gravity.
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Substitute Values Now substitute the values into the equation to find the time of flight: $$ T = \frac{2 \cdot 18.0 \cdot \sin(31^\circ)}{9.81} $$
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Calculate ( \sin(31^\circ) ) Using a calculator or trigonometric tables: $$ \sin(31^\circ) \approx 0.5150 $$
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Complete the Calculation Substituting ( \sin(31^\circ) ): $$ T = \frac{2 \cdot 18.0 \cdot 0.5150}{9.81} $$
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Final Calculation This simplifies to: $$ T = \frac{18.54}{9.81} \approx 1.89 , \text{s} $$
The football hits the ground after approximately ( 1.89 , \text{s} ).
More Information
In projectile motion, the time of flight is dependent on the vertical component of the initial velocity. The angle of projection significantly affects how long the object stays in the air.
Tips
- Not decomposing the initial velocity into horizontal and vertical components.
- Forgetting to use radians when calculating trigonometric functions if the calculator is set to radian mode.
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