Final answer:
The force on the 2-m-tall palm tree is 250 N when hit by a ball. The effective force exerted by the root cannot be calculated with the given information but steps to prevent uprooting include deeper planting, compacted soil, or using stabilizing stakes.
Step-by-step explanation:
Analysis of a Leaning Tree and Physics Calculations
The question involving a tree leaning toward the ground and determining the degrees it has to fall can involve simple geometry, but it is primarily a physics problem concerning forces and moments. For the problem given about the 2-m-tall palm tree, we need to calculate the force on the tree and the effective force by the roots. First, we calculate the change in momentum when the ball hits the tree which is the mass times the change in velocity (Δv). The change in velocity is 5 m/s (the speed of the ball) to 0 m/s after it hits the tree. The time of impact is 10 ms (0.01 s).
(a) The force on the tree (F) is calculated using the impulse-momentum theorem:
F = Δp / Δt = m × Δv / Δt
F = (0.5 kg) × (5 m/s) / (0.01 s) = 250 N
(b) To find the effective force exerted by the root, assume the force is applied at 20 cm from the base. If the tree does not bend, a moment calculation would be needed. However, with the given scenario that the recoil of the tree is minimal, it seems like we do not have enough information to calculate the effective force. Nevertheless, the force can be further discussed as a concept of rooting stability.
(c) To ensure that a tree does not uproot easily, you could have made sure the roots were planted more deeply into the ground, used soil that compacts better, or added a stabilizing stake next to the tree for added support during its initial growth stage.