Chapter 9.2, Problem 24PP

To determine

The velocity of the $4.5\text{kg}$ cart.

Answer to Problem 24PP

  $9\text{cm/s}$ towards right

Explanation of Solution

Given:

The mass of the 1^st cart, $m_1=1.5\text{kg}$

The mass of the 2^nd cart, $m_2=4.5\text{kg}$

Final velocity of 1^st cart when the thread is burned is $v_{1f}=27\text{cm/s}$ towards left.

Formula used:

Momentum of an object of mass m and moving with velocity v is given by,

  $p=mv$

And, law of conservation of linear momentum: Momentum is conserved in a closed and isolated system and is given by,

  $p_i=p_f$

Calculation:

Consider both the carts as a system, then it will be a closed and isolated system and law of conservation of momentum can be applied to it.

Now, since initially both the carts where connected through thread, then initial velocity will be zero,

  $v_{1i}=v_{2i}=0\text{m/s}$

Assume the left direction as negative and right direction as positive, then the final velocity of the 1^st cart can be written as,

  $v_{1f}=-27\text{cm/s}$

Let, the velocity of the 2^nd cart be $v_{2f}$ , then using the law of conservation of linear momentum,

  $\begin{array}{c}{p}_i=p_f\\ m_1{v}_{1i}+m_2{v}_{2i}=m_1{v}_{1f}+m_2{v}_{2f}\end{array}$

Using given values in above,

  $\begin{array}{c}\left(1.5\right)\left(0\right)+\left(4.5\right)\left(0\right)=1.5\left(-27\text{cm/s}\right)+4.5v_{2f}\\ v_{2f}=+9\text{cm/s}\end{array}$

Positive direction denotes right direction.

Conclusion:

Thus, the velocity of the 2^nd cart is $9\text{cm/s}$ towards right.