Canadian star tennis player Bianca Andreescu defeated Serena Williams in the 2019 US Open tournament. During the match Bianca served an ace right up the middle of the court. The ball left her racket 2.69 m above the court with a speed of 50.1 m/s and at an angle of -5.80o. On a tennis court, the net is 11.9 m along the ground from where the ball is served and is 0.914 m high. How far from the net did the ball land on the other side??
Question
Canadian star tennis player Bianca Andreescu defeated Serena Williams in the 2019 US Open tournament. During the match Bianca served an ace right up the middle of the court. The ball left her racket 2.69 m above the court with a speed of 50.1 m/s and at an angle of -5.80o. On a tennis court, the net is 11.9 m along the ground from where the ball is served and is 0.914 m high. How far from the net did the ball land on the other side??
Solution
To solve this problem, we need to break it down into two parts: the horizontal and vertical motion of the ball.
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First, we need to convert the angle from degrees to radians because the trigonometric functions in most programming languages use radians.
-5.80 degrees = -5.80 * π/180 = -0.101 radians
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Next, we calculate the initial velocities in the x and y directions:
Vx = V * cos(θ) = 50.1 m/s * cos(-0.101 rad) = 49.9 m/s Vy = V * sin(θ) = 50.1 m/s * sin(-0.101 rad) = -5.1 m/s
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Now, we can calculate the time it takes for the ball to hit the ground. We can use the equation of motion: y = Vyt + 0.5g*t^2, where y is the height difference, Vy is the initial vertical velocity, g is the acceleration due to gravity (9.81 m/s^2), and t is the time. We can solve this equation for t when y = 2.69 m - 0.914 m = 1.776 m:
1.776 m = -5.1 m/s * t - 0.5 * 9.81 m/s^2 * t^2 Solving this quadratic equation, we find that t = 0.364 s (we discard the negative root because time cannot be negative).
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Finally, we can calculate the horizontal distance the ball travels during this time:
x = Vx * t = 49.9 m/s * 0.364 s = 18.2 m
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The distance from the net where the ball lands is then 18.2 m - 11.9 m = 6.3 m.
So, the ball landed 6.3 m from the net on the other side.
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