a pair of plants has the distance between them decreased by a factor of 5-how do i show that the force between them becomes 25 times as strong?
I mean planets
Gravitational attraction obeys an inverse-square law. If the separation distance goes from x to x/5, the attraction is increased by a factor
[x/(x/5)]^2 = 25
To show that the force between two plants becomes 25 times as strong when the distance between them decreases by a factor of 5, we can use Newton's law of universal gravitation. The equation is as follows:
F = G * (m1 * m2) / r^2
Where:
- F is the force between the two plants
- G is the gravitational constant (approximately 6.67430 x 10^-11 N(m/kg)^2)
- m1 and m2 are the masses of the two plants
- r is the distance between the centers of the two plants
Let's use a variable to represent the initial distance between the two plants: r_initial. Since the distance decreases by a factor of 5, the final distance will be r_final = r_initial/5.
Now, let's denote the initial force as F_initial. According to the question, we want to show that the force becomes 25 times as strong. So, the final force will be F_final = 25 * F_initial.
To compare the forces, we can take the ratio of F_final to F_initial:
(F_final) / (F_initial) = (G * (m1 * m2) / (r_final^2)) / (G * (m1 * m2) / (r_initial^2))
Simplifying the equation, we can cancel out the gravitational constant and the masses:
(F_final) / (F_initial) = (r_initial^2) / (r_final^2)
Plugging in the values, we get:
(F_final) / (F_initial) = (r_initial^2) / ((r_initial/5)^2)
Simplifying further, we have:
(F_final) / (F_initial) = (r_initial^2) / ((r_initial^2) / 25)
(F_final) / (F_initial) = 25
Therefore, the force between the two plants becomes 25 times as strong when the distance between them decreases by a factor of 5.