At the same instant that a 0.5 kg ball is dropped from 25 m above earth , a second ball , with a mass of 0.25 kg , is thrown straight upward from earth’s surface with an initial speed of 15 m/s . They move along nearby lines and pass without colliding. What is the velocity of the center of mass of the two ball system at the end of 2 second?
To find the velocity of the center of mass of the two ball system at the end of 2 seconds, we need to analyze the motion of each ball separately and then calculate the center of mass velocity.
First, let's analyze the motion of the 0.5 kg ball that is dropped from 25 m above Earth's surface. The acceleration due to gravity (g) near the Earth's surface is approximately 9.8 m/s^2. After 2 seconds, the ball would have fallen a distance of (1/2) * g * t^2 = (1/2) * 9.8 * 2^2 = 19.6 m. Since it is a free fall motion, its velocity after 2 seconds would be given by v = g * t = 9.8 * 2 = 19.6 m/s, pointing downwards.
Next, let's analyze the motion of the 0.25 kg ball that is thrown straight upward from Earth's surface with an initial speed of 15 m/s. The acceleration due to gravity acts to oppose the upward motion. After 2 seconds, the ball would have reached its highest point and started to fall back down towards the Earth. The distance traveled upwards would be given by (1/2) * g * t^2 = (1/2) * 9.8 * 2^2 = 19.6 m. However, during the upward motion, the acceleration due to gravity reduces the ball's upward speed. After 2 seconds, the ball's upward velocity would be v = initial velocity - g * t = 15 - 9.8 * 2 = -4.6 m/s, pointing downwards.
Now, we can calculate the center of mass velocity using the principle of conservation of momentum. The momentum of an object is given by the product of its mass and velocity. Since there are no external forces acting on the system of two balls, the total momentum of the system remains constant.
The total momentum of the system can be calculated as follows:
Total momentum = (mass1 * velocity1) + (mass2 * velocity2)
Substituting the values, we get:
Total momentum = (0.5 kg * 19.6 m/s) + (0.25 kg * (-4.6 m/s))
Simplifying, we get:
Total momentum = 9.8 kg * m/s + (-1.15 kg * m/s) = 8.65 kg * m/s
The total mass of the system is 0.5 kg + 0.25 kg = 0.75 kg.
Thus, the velocity of the center of mass of the two ball system at the end of 2 seconds can be calculated as:
Velocity of center of mass = Total momentum / Total mass
Velocity of center of mass = 8.65 kg * m/s / 0.75 kg
Therefore, the velocity of the center of mass of the two ball system at the end of 2 seconds is approximately 11.53 m/s.