To determine the speed of the volleyball when received by the opponent at the same height, we need to consider the horizontal and vertical components of the initial velocity.
Given: Initial speed (vi) = 10 m/s Launch angle (θ) = 40 degrees
First, we'll calculate the horizontal and vertical components of the initial velocity.
Horizontal component (vi_x): vi_x = vi * cos(θ)
Vertical component (vi_y): vi_y = vi * sin(θ)
Now, since the volleyball is received at the same height, the vertical component of its velocity will be the same as the initial velocity at the highest point of its trajectory (when it starts coming down). At this point, the vertical velocity is zero.
Using this information, we can determine the time it takes for the ball to reach the highest point. The formula for the time of flight (t) is:
t = (2 * vi_y) / g
where g is the acceleration due to gravity (approximately 9.8 m/s²).
Next, we can determine the total time of flight, which is twice the time taken to reach the highest point:
total time of flight = 2 * t
Once we have the total time of flight, we can find the horizontal distance traveled by the ball using the formula:
horizontal distance = vi_x * total time of flight
Finally, to find the speed of the ball when received by the opponent, we divide the horizontal distance traveled by the total time of flight:
speed = horizontal distance / total time of flight
Let's calculate it step by step:
vi_x = 10 m/s * cos(40°) vi_x = 10 m/s * 0.766 vi_x ≈ 7.66 m/s
vi_y = 10 m/s * sin(40°) vi_y = 10 m/s * 0.642 vi_y ≈ 6.42 m/s
t = (2 * vi_y) / g t = (2 * 6.42 m/s) / 9.8 m/s² t ≈ 1.31 s
total time of flight = 2 * t total time of flight = 2 * 1.31 s total time of flight ≈ 2.62 s
horizontal distance = vi_x * total time of flight horizontal distance ≈ 7.66 m/s * 2.62 s horizontal distance ≈ 20.09 m
speed = horizontal distance / total time of flight speed ≈ 20.09 m / 2.62 s speed ≈ 7.66 m/s
Therefore, the speed of the ball when received by the opponent at the same height is approximately 7.66 m/s.