Ballistic Pendulum v=0 H M+m M A projectile of mass m moving horizontally with speed v strikes a stationary mass M suspended by strings of length L. Subsequently, m M rise to a height of H Given H, what is the initial speed v of the projectile? Physics 121: Lecture 16, Pg 6
Physics 121: Lecture 16, Pg 6 Ballistic Pendulum A projectile of mass m moving horizontally with speed v strikes a stationary mass M suspended by strings of length L. Subsequently, m + M rise to a height of H. Given H, what is the initial speed v of the projectile? H L L L L m M M + m v V V=0
Ballistic Pendulum Two stage process m collides with M. inelastically. Both M and m then move together with a velocity V(before having risen significantly) 2. M and m rise a height H, conserving energy E (no non-conservative forces acting after collision Physics 121: Lecture 16, Pg 7
Physics 121: Lecture 16, Pg 7 Ballistic Pendulum... Two stage process: 1. m collides with M, inelastically. Both M and m then move together with a velocity V (before having risen significantly). 2. M and m rise a height H, conserving energy E. (no non-conservative forces acting after collision)
Ballistic Pendulum Stage 1: Momentum is conserved m in x-direction: mv=(m+M)V m+M Stage 2: Energy is conserved (EI=EF) 2(m+M)V2=(m+M)gH ) V2=2gH Eliminating v gives m/V<9 Physics 121: Lecture 16, Pg 8
Physics 121: Lecture 16, Pg 8 Ballistic Pendulum... Stage 1: Momentum is conserved in x-direction: mv = (m + M )V V m m M = v + Stage 2: Energy is conserved (E E ) I = F 1 2 2 (m + M )V = (m + M )gH V gH 2 = 2 Eliminating V gives: 2gH m M v 1 = +
Ballistic Pendulum L H M+mi d If we measure the forward displacement d, not H (L-H) L-H H=L-√2-d Hd Physics 121: Lecture 16, Pg 9
Physics 121: Lecture 16, Pg 9 Ballistic Pendulum If we measure the forward displacement d, not H: H L L L L m M M + m v d L H d L-H L d (L H) H L L d 2 2 2 2 2 = + − = − −
Ballistic Pendulum L-H H for M 2gH d m m for d<< l Physics 121: Lecture 16, Pg 10
Physics 121: Lecture 16, Pg 10 Ballistic Pendulum L H d L-H d L for 1 2gH m M v 1 = + L g d m M v 1 = + for d << L