Question

Two gliders are set in motion on an air track. Glider 1 has mass $m_1=0.240kg$and moves to the right with speed 0.740 m/s. It will have a rear-end collision with glider 2, of mass

$m_2=0.360kg$, which initially moves to the right with speed 0.120 m/s. A light spring of force constant 45.0 N/m is attached to the back end of glider 2 as shown in Figure P9.75. When glider 1 touches the spring, superglue instantly and permanently makes it stick to its end of the spring.

(a) Find the common speed the two gliders have when the spring is at maximum compression.

Short answer

(a) The common speed the two gliders$v_a=0.368\text{m}/\text{s}$

Step-by-step solution

Given information

$m_1=0.240kg$

$m_2=0.360kg$

Glider 1speed 0.740 m/s.

Glider 2speed 0.120 m/s

Light spring of force constant 45.0 N/m

Conservation of momentum

Conservation of momentum model states that the momentum of any system will be constant until and unless any external force are getting applied on it.

${(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2})}_{initial}={(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2})}_{final}$

Here,

$\begin{array}{c}{\text{m}}_{\text{1}}=\text{massofspherefirst}\\ \overrightarrow{{\text{v}}_{\text{1}}}=\text{velocityofspherefirst}\\ {\text{m}}_{\text{2}}=\text{massofspheresecond}\\ \overrightarrow{{\text{v}}_{\text{2}}}=\text{velocityofspheresecond}\end{array}$

(a)The common speed the two gliders

If we will consider the first process of spring compression, It continues as long as glider 2 is moving slow than the glider 1. The spring suddenly encounter the maximum compression when the velocity of the both glider will be the same that is $v_a$.

By applying the conservation of the momentum model of the system we will get

$\begin{array}{c}{m}_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}\\ \left(0.240Kg\right)(0.740\text{m}/\text{s})+\left(0.360Kg\right)(0.120\text{m}/\text{s})=\left(0.240Kg\right)v_a+\left(0.360Kg\right)v_a\\ v_a=0.368\text{m}/\text{s}\end{array}$

Suffix i = initial and suffix f = final

The common speed the two gliders is$v_a=0.368\text{m}/\text{s}$