v is the same
forces in ration 1:2
two balls of masses in ratio 1:2 are dropped from the same height find :
a. the ratio between their velocities when they strike the ground and
b. the ratio of the forces acting on them during motion
3 answers
(i) To determine the ratio between the velocities of the two balls when they strike the ground, we can use the principle of conservation of energy. When the balls are dropped from the same height, they both possess potential energy due to their positions, which is converted into kinetic energy as they fall. The conservation of energy can be expressed as:
Potential energy (PE) = Kinetic energy (KE)
Since the potential energy is directly proportional to the mass (m) and the height (h) while the kinetic energy is directly proportional to the mass (m) and the square of the velocity (v), we can write:
m1 * h = 1/2 * m1 * v1^2 (for ball 1)
m2 * h = 1/2 * m2 * v2^2 (for ball 2)
Given that the mass ratio between the balls is 1:2, we can express m2 in terms of m1 as m2 = 2 * m1:
m1 * h = 1/2 * m1 * v1^2
2 * m1 * h = 1/2 * 2^2 * m1 * v2^2
2 * h = v2^2
Taking the square root of both sides, we get:
sqrt(2 * h) = v2
Therefore, the ratio of velocities when the balls strike the ground is:
v2/v1 = sqrt(2 * h) / v1
(ii) The ratio of the forces acting on the balls during motion can be determined using Newton's second law of motion, which states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a):
F = m * a
Since the balls are dropped from the same height, they experience the same acceleration due to gravity (g). Therefore, the ratio of forces acting on the balls can be expressed as:
F2/F1 = (m2 * a) / (m1 * a)
Since the acceleration (a) cancels out, we can simplify the expression:
F2/F1 = m2/m1 = (2 * m1) / m1 = 2
Therefore, the ratio of the forces acting on the balls during motion is 2:1.
Potential energy (PE) = Kinetic energy (KE)
Since the potential energy is directly proportional to the mass (m) and the height (h) while the kinetic energy is directly proportional to the mass (m) and the square of the velocity (v), we can write:
m1 * h = 1/2 * m1 * v1^2 (for ball 1)
m2 * h = 1/2 * m2 * v2^2 (for ball 2)
Given that the mass ratio between the balls is 1:2, we can express m2 in terms of m1 as m2 = 2 * m1:
m1 * h = 1/2 * m1 * v1^2
2 * m1 * h = 1/2 * 2^2 * m1 * v2^2
2 * h = v2^2
Taking the square root of both sides, we get:
sqrt(2 * h) = v2
Therefore, the ratio of velocities when the balls strike the ground is:
v2/v1 = sqrt(2 * h) / v1
(ii) The ratio of the forces acting on the balls during motion can be determined using Newton's second law of motion, which states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a):
F = m * a
Since the balls are dropped from the same height, they experience the same acceleration due to gravity (g). Therefore, the ratio of forces acting on the balls can be expressed as:
F2/F1 = (m2 * a) / (m1 * a)
Since the acceleration (a) cancels out, we can simplify the expression:
F2/F1 = m2/m1 = (2 * m1) / m1 = 2
Therefore, the ratio of the forces acting on the balls during motion is 2:1.
Note: It is assumed that the air resistance is negligible in this scenario.