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 certain speed and angle to hit the ground, which involves understanding projectile motion.
Answer
The time taken for the football to hit the ground is approximately $1.89 \, s$.
Answer for screen readers
The time it takes for the football to hit the ground is approximately $1.89 , s$.
Steps to Solve
-
Identify Initial Velocity Components To determine the time of flight, we first need to break down the initial velocity into its horizontal ($v_{x}$) and vertical ($v_{y}$) components using the angle $\theta = 31°$ and speed $v_0 = 18.0 , m/s$.
- The horizontal component: $$ v_{x} = v_0 \cdot \cos(\theta) = 18.0 \cdot \cos(31°) $$
- The vertical component: $$ v_{y} = v_0 \cdot \sin(\theta) = 18.0 \cdot \sin(31°) $$
-
Calculate Vertical Velocity Component Calculate the vertical component: $$ v_{y} = 18.0 \cdot \sin(31°) \approx 18.0 \cdot 0.5150 \approx 9.27 , m/s $$
-
Apply the Time of Flight Formula The time of flight for a projectile launched and landing at the same height can be found using the formula: $$ t = \frac{2 v_{y}}{g} $$ where $g = 9.81 , m/s^2$ is the acceleration due to gravity.
-
Substitute Values into the Formula Substituting the value of $v_{y}$: $$ t = \frac{2 \cdot 9.27}{9.81} \approx \frac{18.54}{9.81} \approx 1.89 , s $$
The time it takes for the football to hit the ground is approximately $1.89 , s$.
More Information
Projectile motion is influenced by the initial speed and the angle of launch. The time of flight depends primarily on the vertical component of the initial velocity and the acceleration due to gravity.
Tips
- Neglecting the effect of gravity: Ensure to use the gravitational acceleration when calculating time of flight.
- Confusing horizontal and vertical motion: Remember that horizontal motion does not affect the time of flight in this scenario.
AI-generated content may contain errors. Please verify critical information