Question

Object $\mathbf{A}$ has mass ${\mathbf{m}}_{\mathbf{A}}\mathbf{=}\mathbf{7}\mathbf{}\mathbf{kg}$and initial momentum${\overrightarrow{\mathbf{P}}}_{\mathbf{Aj}}\mathbf{=}\mathbf{(}\mathbf{17}\mathbf{,}\mathbf{-}\mathbf{5}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{kg}\mathbf{.}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$, just before it strikes object B , which has mass ${\mathbf{m}}_{\mathbf{A}}\mathbf{=}\mathbf{11}\mathbf{kg}$. Object B has initial momentum ${\overrightarrow{\mathbf{p}}}_{\mathbf{Bj}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{kh}\mathbf{.}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. After the collision, object A is observed to have final momentum ${\overrightarrow{\mathbf{P}}}_{\mathbf{Af}}\mathbf{=}\mathbf{(}\mathbf{13}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{kg}\mathbf{.}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. In the following questions, “initial” refers to values before the collisions, and “final” refers to values after the collision. Consider a system consisting of both objects and . Calculate the following quantities: (a) The total initial momentum of this system. (b) The final momentum of object B. (c) The initial kinetic energy of object A. (d) The initial kinetic energy of object B. (e) The final kinetic energy of object A. (f) The final kinetic energy of object B. (g) The total initial kinetic energy of the system. (h) The total final kinetic energy of the system. (i) The increase of internal energy of the two objects. (j) What assumption did you make about Q (energy flow from surroundings into the system due to a temperature difference)?

Short answer

(a) (21,1,0) kg.m/s

(b) (8,-2,0) kg.m/s

(c) (20.64,1.78,0) m/s

(d) (0.727,1.636,0) m/s

(e) (12.07,0.64,0) m/s

(f) (2.909,0.64,0) m/s

(g) (21.367,3.416,0) m/s

(h) (14.979,0.821,0) m/s

(i) (8.57,1.14,0) m/s and (-2.182,1.455,0) m/s

(j) 0

Step-by-step solution

Identification of the given data

The given data is listed below as,

• The mass of the object A is, $m_A=7\mathrm{kg}$ .
• The initial momentum of object A is, localid="1657860376383" ${\overrightarrow{P}}_{A,i}=(17,-5,0)kg.m/s$ .
• The mass of object B is, $m_B=11kg$.
• The initial momentum of object B is, ${\overrightarrow{P}}_{B,i}=(4,6,0)k.m/s$ .
• After the collision, the final momentum of object A is,${\overrightarrow{P}}_{A,f}=(13,3,0)kg.m/s.$ ${\overrightarrow{P}}_{A,f}=(13,3,0)kg.m/s$.

Significance of the law of conservation of momentum, the first law of thermodynamics, and work-energy theorem for the objects

This law of conservation of momentum states that the total momentum of an object before and after collision with another object becomes equal.

The first law of thermodynamics states that the change in the internal energy of a system is equal to the difference between the heat added and the work done by the system.

The work-energy theorem states that the net change in the work is equal to the change in the kinetic energy.

The law of conservation of momentum gives the initial and the final momentum of objects A, B, and the system. The first law of thermodynamics and the work-energy theorem gives the increase in the internal energy of the system along with its kinetic energies of them.

Determination of the initial momentum

The expression for the total initial momentum of the system is expressed as,

${\overrightarrow{P}}_{initial}={\overrightarrow{P}}_{A,i}+{\overrightarrow{P}}_{B,i}$

For${\overrightarrow{P}}_{A,i}=(17,-5,0)kg.m/s$ and${\overrightarrow{P}}_{B,i}=(4,6,0)kg.m/s$ ,

$$\begin{gathered} {\overrightarrow{P}}_{initial}=(17,-5,0)kg.m/s+(4,6,0)kg.m/s \\ =(21,1,0)kg.m/s \end{gathered}$$

Thus, the total initial momentum of the system is $(21,1,0)kg.m/s$.

Determination of the final momentum of object B

The expression of the final momentum is expressed as,

${\overrightarrow{P}}_{initial}={\overrightarrow{P}}_{A,i}+{\overrightarrow{P}}_{B,i}$

As role="math" localid="1657861309005" ${\overrightarrow{P}}_{initial}={\overrightarrow{P}}_{initial}$ , then the above equation can be written as,

${\overrightarrow{P}}_{initial}={\overrightarrow{P}}_{A,i}+{\overrightarrow{P}}_{B,i}$

For${\overrightarrow{P}}_{initial}=(21,1,0)kg.m/s$and${\overrightarrow{P}}_{A,i}=(13,3,0)kg.m/s$,

$$\begin{gathered} (21,1,0)kg.m/s=(13,3,0)kg.m/s+{\overrightarrow{P}}_{B,i} \\ {\overrightarrow{P}}_{B,i}=(8,-2,0)kg.m/s \end{gathered}$$

Thus, the final momentum of object B is $(8,-2,0)kg.m/s$.

Determination of the initial kinetic energy of object A

The equation of the initial kinetic energy of object A is expressed as,

$K.E_{.A,initial}=\frac{{\overrightarrow{P}}_{A,i^2}}{2m_A}$

Here, $p_{A,i}$ is the initial momentum of object A and $m_A$ is the mass of object A.

For${\overrightarrow{P}}_{A,i}=(17,-5,0)kg.m/s$and $m_A=7kg$,

$$\begin{gathered} K.E_{.A,initial}=\frac{{\left((17,-5,0)kg.m/s\right)}^2}{2\times 7kg} \\ =\frac{\left((289,25,0)kg.m/s\right)}{14kg} \\ =(20.64,1.78,0)m/s \end{gathered}$$

Thus, the initial kinetic energy of object A is $(20.64,1.78,0)m/s$.

Determination of the initial kinetic energy of object B

$p_{B,i}$The equation of the initial kinetic energy of the object B is expressed as,

localid="1657863580396" $K.E_{.B,initial}=\frac{{\overrightarrow{P}}_{B,i^2}}{2m_B}$

Here, $p_{B,i}$ is the initial momentum of object B and $m_B$ is the mass of object B.

For ${\overrightarrow{p}}_{B,i}=(4,6,0)kg.m/s$and $m_B=11kg$,

localid="1657863591107" $$\begin{gathered} K.E_{.B,initial}=\frac{{\left((4,6,0)kg.m/s\right)}^2}{2\times 11kg} \\ =\frac{\left((16,36,0)kg.m/s\right)}{22kg} \\ =(0.727,1.636,0)m/s \end{gathered}$$

Thus, the initial kinetic energy of object B is $(0.727,1.636,0)m/s$.

Determination of the final kinetic energy of object A

The equation of the final kinetic energy of the object A is expressed as,

$K.E_{.A,final}=\frac{{\overrightarrow{P}}_{A,i^2}}{2m_A}$

Here, ${\overrightarrow{P}}_{A,i}$ is object A’s initial momentum is the mass of object A.

For${\overrightarrow{p}}_{A,I}=(13,3,0)kg.m/s$and $m_A=7kg$,

localid="1657863624578" $$\begin{gathered} K.E_{.A,final}=\frac{{\left((13,3,0)kg.m/s\right)}^2}{2\times 7kg} \\ =\frac{\left((169,9,0)kg.m/s\right)}{14kg} \\ =(12.07,0.64,0)m/s \end{gathered}$$

Thus, the final kinetic energy of object A is $(12.07,0.64,0)m/s$.

Determination of the final kinetic energy of object B

The equation of the final kinetic energy of object B is expressed as,

$K.E_{.B,final}=\frac{{\overrightarrow{P}}_{B,i^2}}{2m_B}$

Here, ${\overrightarrow{P}}_{B,i}$ is object B’s initial momentum is the mass of object B.

For${\overrightarrow{P}}_{B,i}=(8,-2,0)kg.m/s$and$m_B=11kg$,

localid="1657864819624" $$\begin{gathered} K.E_{B,final}=\frac{{\left((8,-2,0)kg.m/s\right)}^2}{2\times 11kg} \\ =\frac{{\left((64,4,0)kg.m/s\right)}^2}{22kg} \\ =(2.909,0.181,0)m/s \end{gathered}$$

Thus, the final kinetic energy of object B is $(2.909,0.181,0)m/s$.

Determination of the total initial kinetic energy of the system

The equation of the total initial kinetic energy of the system is expressed as,

$K.E_{.initial}=K.E_{.A,initial}+K.E_{.B,initial}$

For$K.E_{.A,initial}=(20.64,1.78,0)m/s$and$K.E_{.B,initial}=(0.727,1.636,0)m/s.$,

$$\begin{gathered} K.E_{initial}=(20.64,1.78,0)m/s+(0.727,1.636,0)m/s \\ =(21.367,3.416,0)m/s \end{gathered}$$

Thus, the total initial kinetic energy of the system is $(21.367,3.416,0)m/s$.

Determination of the total final kinetic energy of the system

The equation of the total final kinetic energy of the system is expressed as,

$K.E_{.final}=K.E_{.A,final}+K.E_{.B,final}$

Forrole="math" localid="1657865760801" $K.E_{.A,final}=(12.07,0.64,0)m/s$and$K.E_{.B,final}=(12.07,0.64,0)m/s$,

$$\begin{gathered} K.E_{.A,final}=(12.07,0.64,0)m/s+(2.909,0.181,0)m/s \\ =(14.979,0.821,0)m/s \end{gathered}$$

Thus, the total final kinetic energy of the system is $(14.979,0.821,0)m/s$.

Determination of the increase in the internal energy of the objects

From the first law of thermodynamics, the equation of the change in the internal energy can be expressed as,

$∆l=∆Q-∆W$

Here, $∆l$is the change in the system’s internal energy, $∆Q$is the change in the energy flow, and $∆W$is the work done by the objects.

For the object A,

Rewrite the above equation by considering the system as an isolated system; that is, the value of the energy flow is 0 $∆Q=0$ and $∆W=(∆k.E_{.A,final}-∆k.E_{.B,initial})$.

role="math" localid="1657866325087" $∆l=0-(∆k.E_{.A,final}-∆k.E_{.A,initial})$

For$∆K.E_{.A,final}=(12.07,0.64,0)m/s$and$∆k.E_{.A,initial}=(20.64,1.78,0)m/s$,

$$\begin{gathered} ∆l=0-\left((12.07,0.64,0m/s-(20.64,1.78,0)m/s\right) \\ =(8.57,1.14,0)m/s \end{gathered}$$

For the object B,

The expression for the change in internal energy is as follows,

$∆l=0-(∆K.E_{B,final}-∆K.E_{B,initial})$

For$∆K.E_{B,final}=(2.909,0.181,0)m/s$and$∆K.E_{B,initial}=(0.727,1.636,0)m/s$,

$$\begin{gathered} ∆l=0-\left((2.909,0.181,0)m/s-(0.727,1.636,0)m/s\right) \\ =(-2.182,1.455,0)m/s \end{gathered}$$

Thus, the increase of internal energy of object A is $(8.57,1.14,0)m/s$and of object B is $(-2.182,1.455,0)m/s$.

Determination of the assumption of energy flow

The system is assumed to be an isolated system. So, the value of energy flow from the surroundings to the system is zero.

Thus, Q is assumed to be 0.