A football is kicked from the ground at an angle 30degree above the horizontal with an initial speed 18m/s and directly towards the goalkeeper. If the goal line is a horizontal distance 25m away calculate the vertical height at which the ball crosses the goal line if not caught by the goalkeeper take g = 10m/s'2
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To calculate the vertical height at which the ball crosses the goal line, we need to break down the motion of the football into its horizontal and vertical components.
Given:
- Initial speed of the football (u) = 18 m/s
- Angle of projection (θ) = 30 degrees
- Horizontal distance to the goal line (x) = 25 m
- Acceleration due to gravity (g) = 10 m/s^2
1. Break down the initial velocity into horizontal and vertical components:
- Horizontal component: ucosθ (cosine of the angle)
- Vertical component: usinθ (sine of the angle)
2. Calculate the time taken for the ball to reach the goal line (t):
- We know that horizontal distance (x) = (horizontal component) * t
- Substitute the values: 25 = (ucosθ) * t
3. Solve for t:
- Rearrange the equation to isolate t: t = 25 / (ucosθ)
4. Calculate the vertical distance (y) at time t:
- We know that vertical distance (y) = (vertical component) * t - (1/2)gt^2
- Substitute the values: y = (usinθ) * t - (1/2)gt^2
5. Substituting the value of t obtained in step 3:
- Substitute t = 25 / (ucosθ) in the equation: y = (usinθ) * (25 / (ucosθ)) - (1/2)g(25^2 / (u^2cos^2θ))
6. Simplify the equation:
- Cancel out u, sinθ, and cosθ terms: y = 25tanθ - (1/2)g(25^2 / (cos^2θ))
7. Calculate the value of y:
- Substitute the given values: y = 25 * tan(30) - (1/2) * 10 * (25^2 / cos^2(30))
Thus, by calculating the value of y, you will determine the vertical height at which the ball crosses the goal line if not caught by the goalkeeper.