A raindrop reaching the ground with terminal velocity has momentum P. Another drop of twice the radius, also reaching the ground with terminal velocity, will have momentum.
Question
A raindrop reaching the ground with terminal velocity has momentum P. Another drop of twice the radius, also reaching the ground with terminal velocity, will have momentum.
Solution
The momentum of an object is given by the product of its mass and velocity (P = mv).
In the case of a raindrop falling at terminal velocity, the forces acting on it are balanced, so the net force is zero. The forces acting on the raindrop are gravity (downwards) and air resistance (upwards).
The gravitational force is proportional to the volume of the raindrop (which is proportional to the cube of the radius, r^3), and the air resistance is proportional to the cross-sectional area of the raindrop (which is proportional to the square of the radius, r^2).
So, if we double the radius of the raindrop, the gravitational force (and hence the mass of the raindrop) increases by a factor of 2^3 = 8, and the air resistance (and hence the terminal velocity of the raindrop) decreases by a factor of 2^2 = 4.
Therefore, the momentum of the larger raindrop (P') is given by P' = m'v' = (8m)(v/4) = 2P.
So, a raindrop of twice the radius, also reaching the ground with terminal velocity, will have twice the momentum of the original raindrop.
Similar Questions
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