To find the horizontal distance traveled by the baseball before reaching its original level, we can analyze the projectile motion of the ball.
Given: Initial velocity (v0) = 100 m/s Launch angle (θ) = 30°
First, we need to split the initial velocity into its horizontal and vertical components. The horizontal component (v0x) remains constant throughout the motion, while the vertical component (v0y) changes due to gravity.
The horizontal component of velocity is given by: v0x = v0 * cos(θ)
The vertical component of velocity is given by: v0y = v0 * sin(θ)
To determine the time it takes for the ball to reach its original level, we can use the equation for vertical displacement (Δy): Δy = v0y * t + (1/2) * g * t^2
Since the ball reaches its original level, the vertical displacement is zero (Δy = 0), and we can solve for time (t).
0 = v0y * t + (1/2) * g * t^2
Since the ball is at its original level, the time to reach the maximum height and return to the original level is twice the time calculated above. Let's call this time T.
T = 2 * t
Finally, we can calculate the horizontal distance traveled (d) using the equation: d = v0x * T
Substituting the expressions for v0x and T, we have: d = v0 * cos(θ) * 2 * t
Now, we need to solve for t.
From the equation for vertical displacement, we have: 0 = v0y * t + (1/2) * g * t^2
Since v0y = v0 * sin(θ), we can rewrite the equation as: 0 = v0 * sin(θ) * t + (1/2) * g * t^2
Rearranging the equation, we get: (1/2) * g * t^2 = -v0 * sin(θ) * t
Simplifying further: (1/2) * g * t = -v0 * sin(θ)
Solving for t: t = -2 * v0 * sin(θ) / g
Now that we have the value of t, we can substitute it back into the equation for horizontal distance (d):
d = v0 * cos(θ) * 2 * t
Plugging in the given values:
v0 = 100 m/s θ = 30° g = 9.8 m/s^2 (acceleration due to gravity)
First, let's calculate t:
t = -2 * 100 * sin(30°) / 9.8
Using the trigonometric value for sin(30°) = 0.5:
t = -2 * 100 * 0.5 / 9.8
t ≈ -10.2041 s
Since time cannot be negative in this context, we consider the positive value:
t ≈ 10.2041 s
Now we can calculate the horizontal distance:
d = 100 * cos(30°) * 2 * 10.2041
Using the trigonometric value for cos(30°) = √3/2:
d = 100 * (√3/2) * 2 * 10.2041
d ≈ 1758.43 meters
Therefore, the baseball will attain its original level approximately 1758.43 meters away from the throwing point.