Understanding the Problem
* Initial Conditions: We know the ball starts at a height of 9.1 meters and has an initial velocity of (7.6i + 6.1j) m/s.
* Goal: We need to find the maximum height the ball reaches.
Physics Concepts
* Projectile Motion: The ball is undergoing projectile motion, which means its motion can be analyzed in the horizontal (x) and vertical (y) directions separately.
* Vertical Motion: The vertical motion is influenced by gravity, causing the ball to slow down as it goes up and then speed up as it falls back down.
* Maximum Height: At the maximum height, the ball's vertical velocity is zero.
Solution
1. Vertical Velocity Component: The initial vertical velocity is the y-component of the initial velocity vector: v_iy = 6.1 m/s.
2. Acceleration due to Gravity: The acceleration due to gravity is acting downward: a_y = -9.8 m/s².
3. Vertical Motion Equation: We can use the following kinematic equation to find the maximum height (h):
v_fy² = v_iy² + 2a_y(h - h_i)
* v_fy = final vertical velocity (0 m/s at the maximum height)
* v_iy = initial vertical velocity (6.1 m/s)
* a_y = acceleration due to gravity (-9.8 m/s²)
* h = maximum height (what we want to find)
* h_i = initial height (9.1 m)
4. Solving for h:
0² = 6.1² + 2(-9.8)(h - 9.1)
37.21 = 19.6(h - 9.1)
h - 9.1 = 1.9
h = 11 m
Answer
The maximum height reached by the ball is 11 meters.
