Final answer:
The maximum height a pebble reaches when thrown is directly proportional to the square of its initial velocity. For an initial speed of 10 m/s, it achieved a height of 5 m, which helped determine the proportionality constant. Using this, we calculated the maximum height for an initial speed of 12 m/s to be 7.2 m and the initial speed needed to reach a maximum height of 16 m to be approximately 17.89 m/s.
Step-by-step explanation:
The question involves the concept of projectile motion and how the maximum height (H) reached by a projectile (pebble in this case) is directly proportional to the square of its initial velocity (U^2). According to the given information, when the pebble is thrown with an initial speed of 10 m/s, it reaches 5 m height. Using this as our base reference, we can establish the constant of proportionality. Let's denote this constant as k, so we can write the relationship as H = k × U^2.
For the initial speed of 10 m/s and height of 5 m, we get 5 = k × (10)^2. Solving for k, we find that k = 0.05. Now, we can use this constant to calculate the maximum heights for other initial speeds. For an initial speed of 12 m/s, applying the relationship, we get H = 0.05 × (12)^2, which equals 7.2 m. For the given maximum height of 16 m, we want to find the initial speed. We rearrange the equation to solve for U: U = √(H/k) = √(16/0.05), which yields U approximately 17.89 m/s. Therefore, the initial speed required to reach 16 m height is about 17.89 m/s.