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Answer :
This question involves concepts related to projectile motion in physics, specifically involving a rugby ball kicked at an angle. Let's break down each part:
4.1. Determine the x and y components of the initial velocity.
To find the components of the initial velocity, we use trigonometric functions:
The x-component (horizontal) of the velocity is given by:
[tex]v_x = v \cdot \cos(\theta)[/tex]
where [tex]v = 60.0 \, \text{m/s}[/tex] and [tex]\theta = 28^\circ[/tex].[tex]v_x = 60.0 \times \cos(28^\circ) \approx 60.0 \times 0.8829 \approx 52.97 \, \text{m/s}[/tex]
The y-component (vertical) of the velocity is given by:
[tex]v_y = v \cdot \sin(\theta)[/tex][tex]v_y = 60.0 \times \sin(28^\circ) \approx 60.0 \times 0.4695 \approx 28.17 \, \text{m/s}[/tex]
4.2. Calculate the maximum height reached by the ball.
The formula for the maximum height [tex]h_{max}[/tex] in projectile motion, assuming upward velocity only, is:
[tex]h_{max} = \frac{{v_y^2}}{{2g}}[/tex]
where [tex]g = 9.81 \, \text{m/s}^2[/tex] is the acceleration due to gravity.
[tex]h_{max} = \frac{{(28.17)^2}}{{2 \times 9.81}} \approx \frac{{793.0689}}{{19.62}} \approx 40.42 \, \text{m}[/tex]
4.3. Calculate the time taken by the ball to reach maximum height.
The time to reach the maximum height [tex]t_{max}[/tex] can be calculated by:
[tex]t_{max} = \frac{{v_y}}{{g}}[/tex]
[tex]t_{max} = \frac{{28.17}}{{9.81}} \approx 2.87 \, \text{seconds}[/tex]
4.4. Calculate the time of travel before the ball hits the ground.
The total time of flight [tex]T[/tex] for a projectile launched from and landing on the same horizontal level is:
[tex]T = 2 \times t_{max}[/tex]
[tex]T = 2 \times 2.87 \approx 5.74 \, \text{seconds}[/tex]
4.5. Calculate how far away the ball hits the ground.
The horizontal distance [tex]R[/tex] (also called the range) is given by:
[tex]R = v_x \times T[/tex]
[tex]R = 52.97 \times 5.74 \approx 304.044 \, \text{m}[/tex]
These calculations show how the components of the initial velocity are used to determine other characteristics of projectile motion, like maximum height, time, and range.
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