To solve this problem, we need to analyze the motion of a particle that is gaining mass while being projected vertically upwards. The key elements here are the initial speed, the changing mass, and the forces acting on the particle as it ascends. Let's break it down step by step.
Understanding the Motion of the Particle
Initially, the particle has a mass of m and is projected upwards with a speed of gT, where g is the acceleration due to gravity and T is a constant. As the particle moves upwards, its mass increases according to the equation:
m(t) = me^(t/T)
This indicates that at time t, the mass of the particle is me^(t/T). The added mass is at rest relative to the particle when it is acquired, meaning that it does not contribute to the initial velocity of the particle.
Applying Newton's Second Law
To find the time when the particle reaches its highest point, we can apply Newton's second law, which states that the net force acting on an object is equal to the rate of change of momentum. The forces acting on the particle are:
- The gravitational force acting downwards: F_gravity = m(t)g = me^(t/T)g
- The upward thrust due to the initial velocity and the mass increase.
At the highest point, the velocity of the particle becomes zero. We can express the momentum of the particle as:
P(t) = m(t)v(t) = me^(t/T)(gT - gt)
Setting the velocity to zero at the highest point gives us:
gT - gt = 0
Finding the Time at the Highest Point
From the equation above, we can solve for t:
gT = gt
Dividing both sides by g (assuming g ≠ 0):
T = t
This means that the particle reaches its highest point at time t = T.
Calculating the Mass at the Highest Point
Now that we have determined the time at which the particle reaches its highest point, we can find the mass at that instant:
m(T) = me^(T/T) = me^1 = me
Summary of Results
In summary, the particle reaches its highest point at time T, and at that instant, its mass is me. This analysis illustrates how the changing mass of the particle affects its motion and highlights the interplay between mass, velocity, and gravitational forces in a dynamic system.
