To determine the initial velocity of the ball, we can use the kinematic equation that relates the initial velocity (v₀), final velocity (v), acceleration (a), and displacement (d). Assuming no air resistance, we can use the equation:
v² = v₀² + 2ad
In this case, the ball is thrown upward to a height of 3 meters, which means the displacement (d) is +3 meters (taking upward as the positive direction). Since the ball reaches its highest point at the top of its trajectory, the final velocity (v) at that point is 0 m/s.
Plugging in these values into the equation, we have:
0² = v₀² + 2(-9.8 m/s²)(3 m)
Simplifying the equation:
0 = v₀² - 58.8 m²/s²
Rearranging the equation:
v₀² = 58.8 m²/s²
Taking the square root of both sides to solve for v₀:
v₀ = ±√(58.8 m²/s²)
Therefore, the initial velocity of the ball is approximately ±7.67 m/s. The positive value represents the upward initial velocity, while the negative value represents the downward initial velocity.