Final answer:
The boy should take his stand about 12 m from the wall to make the ball clear the roof with the least effort.
Step-by-step explanation:
To determine how far from the wall the boy should take his stand to make the ball clear the roof with the least effort, we can use the concepts of projectile motion. First, we need to find the time it takes for the ball to travel horizontally a distance of 12 m. Using the formula d = v*t, where d is the distance, v is the horizontal component of the velocity, and t is the time, we can rearrange the formula to solve for t: t = d / v. Since the ball is thrown horizontally, the horizontal component of the velocity is the same as the initial horizontal velocity. From problem 27, we know that the horizontal distance is 100 m. So, the time it takes for the ball to travel horizontally 12 m is t = (12 m) / (v).
Next, we can use the formula for vertical motion to find the time it takes for the ball to travel vertically a height of 7.5 m. Using the formula h = 1/2 * g * t^2, where h is the height, g is the acceleration due to gravity, and t is the time, we can rearrange the formula to solve for t: t = √(2h/g). Plugging in the values, we have t = √(2*7.5 m / 9.8 m/s^2) = √1.53 s.
Since the total time for the ball to travel horizontally 12 m and vertically 7.5 m is the same as the time taken for the ball to travel horizontally 100 m, we can equate the two equations and solve for v: (12 m) / v = 1.53 s. Rearranging the equation, we have v = (12 m) / 1.53 s = 7.84 m/s. Finally, we can calculate the distance the boy should take his stand from the wall by multiplying the horizontal component of the velocity by the time taken to travel horizontally 12 m: d = v * t = (7.84 m/s) * (1.53 s) = 11.98 m ≈ 12 m. Therefore, the boy should take his stand about 12 m from the wall to make the ball clear the roof with the least effort.