A tennis player serves a ball with a speed of 30m/s.The ball leaves the racquet at a height of 2.5 and a horizontal distance of 12.25m from the net.The height of the net is 0.91m
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To solve this problem, we can break it down into two components: the horizontal and vertical motions of the ball.
1. Horizontal Motion:
The horizontal distance traveled by the ball is given as 12.25m. We can calculate the time taken by the ball to travel this distance using the formula:
Distance = Speed * Time
Rearranging the formula, we get:
Time = Distance / Speed
Substituting the values, we have:
Time = 12.25m / 30m/s
Time = 0.4083 seconds
2. Vertical Motion:
The ball leaves the racquet at a height of 2.5m and the height of the net is 0.91m. We can calculate the time taken for the ball to reach the net height using the equation of motion:
Final Height = Initial Height + (Initial Vertical Velocity * Time) + (0.5 * Acceleration * Time^2)
Since the ball is served horizontally, the initial vertical velocity is 0. We can rearrange the equation to solve for time:
Time = sqrt((Final Height - Initial Height) / (0.5 * Acceleration))
Substituting the values, we have:
Time = sqrt((0.91m - 2.5m) / (0.5 * (-9.8m/s^2)))
Time = sqrt(-1.59m / -4.9m/s^2)
Time = sqrt(0.3245s^2)
Time = 0.57 seconds
Therefore, the time taken for the ball to reach the net height is 0.57 seconds.
In summary:
- The time taken for the ball to reach the horizontal distance of 12.25m is 0.4083 seconds.
- The time taken for the ball to reach the net height is 0.57 seconds.
To find out if the ball clears the net, we need to compare the height at which the ball leaves the racquet with the height of the net.
First, let's calculate the time it takes for the ball to travel the horizontal distance of 12.25m.
We can use the formula: distance = speed x time.
Rearranging the formula, we get: time = distance / speed.
Plugging in the values, we have: time = 12.25m / 30m/s = 0.4083s (rounded to 4 decimal places).
Now, let's calculate the vertical distance the ball traveled during that time.
We can use the formula: distance = initial velocity x time + 0.5 x acceleration x time^2.
The initial vertical velocity is 0 since the ball leaves the racquet horizontally. The acceleration due to gravity is approximately -9.8m/s^2, taking downward as the negative direction.
The formula becomes: distance = 0 x time + 0.5 x (-9.8m/s^2) x time^2.
Simplifying, we have: distance = -4.9 x time^2.
Plugging in the value for time calculated earlier, we get: distance = -4.9 x (0.4083s)^2 = -0.825m (rounded to 3 decimal places).
Since the height of the net is 0.91m, and the ball traveled -0.825m in the vertical direction, the ball does not clear the net.