Question

(3) A body moving so that,at time t,it's mass is m, it's velocity is v and the resultant force acting on the body is F. Suppose that at this instant, a particle of mass om and velocity u coaleces with the body so that, at time ot, the body has the mass m + Sm and the velocity v + öv. Prove that Fma (v- D-u a. A body with initial mass rn is projected vertically upwards with speed where k is a constant. At any subsequent time t its speed is v and its mass is mek, The particles of added mass were all at rest when picked up by the body. Find the mass of the body at its highest point. b.A particle is projected vertically upward with speed u in a resistive medium at point A The resistive force on the particle IS mkv2 where v is the velocity of the particle and k is a constant. Prove that the maximum height attained by the particle is In(1 + ^u^). If the particle will return to the point A with velocity au prove that 1+ 2k ku2

Answer 1

Using impulse momentum method F dt = m_f v_f - m_i v_i = p_F - p_i p_f = (m + delta m)(v + delta v) = mv + m delta v + v delta m + delta m delta v^0 = mv + m dv + v dm p_i = mv + 48m F dt = mv + m dv + v dm - mv - u dm F = m dv/dt + (v - u) dm/dt -(1) using eqn (1): given that u = 0 (colloidal particle are rest) F = mg = me^kt g mg = m dv/dt + v dm/dt implies m = m_0 e^kt: dm/dt = m_0 k e^kt implies m_0 e^kt g = m_0 ekt dv/dt + v m_0 ke^kt \r\n g = dv/dt + vk g - vk = dv/dt implies integral^0_v_0 dv/g - v_k = integral^t_0 dt implies ln (g/g - v_0k) = t implies so mass at time 't'