Question

A mass of 200 gm has initial velocity \[{{V}_{i}}=2i+3j\]and final velocity \[{{V}_{f}}=-2i-3j\]. Find the magnitude of change in momentum
\[A.\,|\Delta \overset{\to }{\mathop{p}}\,|=3.84\]
\[B.\,|\Delta \overset{\to }{\mathop{p}}\,|=3.04\]
\[C.\,|\Delta \overset{\to }{\mathop{p}}\,|=1.44\]
\[D.\,|\Delta \overset{\to }{\mathop{p}}\,|=2.04\]

Answer 1

Hint: Newton's second law of motion states that "The rate of change of momentum is equal to the external force acting on a body." The amount of change in an object's momentum is called the impulse. So, the change in momentum is the product of mass and change in its velocity.
Formula used:
\[\Delta p=m\times ({{V}_{f}}-{{V}_{i}})\]

Complete answer:
From given, we have,
The mass = 200 gm
Therefore, the mass = 0.2 kg
The initial velocity of the mass, \[{{V}_{i}}=2i+3j\]
The final velocity of the mass, \[{{V}_{f}}=-2i-3j\]
The change in the momentum, \[\Delta p\]= ?
The change in momentum is given by the formula,
\[F\times \Delta t=\Delta p\]
Where F is the force and \[\Delta t\]is the change in time.
Further this formula can be expressed in terms of mass and the velocity.
\[\begin{align}
  & \Delta p=m{{V}_{f}}-m{{V}_{i}} \\
 & \Delta p=m({{V}_{f}}-{{V}_{i}}) \\
\end{align}\]
Where m is the mass, \[({{V}_{f}},{{V}_{i}})\]are the final and initial velocities respectively.
As the given values are in vector form, we will first find the velocity change, that is, the difference between the final velocity and the initial velocity.
As the vector subtraction is not possible, we will use vector addition to solve.
\[\begin{align}
  & {{V}_{f}}-{{V}_{i}}={{V}_{f}}+(-{{V}_{i}}) \\
 & \Rightarrow {{V}_{f}}+(-{{V}_{i}})=(-2i-3j)+[-(2i+3j)] \\
 & \Rightarrow {{V}_{f}}+(-{{V}_{i}})=-2i-3j-2i-3j \\
 & \Rightarrow {{V}_{f}}-{{V}_{i}}=-4i-6j \\
\end{align}\]
Therefore, the change in the velocity is, \[{{V}_{f}}-{{V}_{i}}=-4i-6j\]
Now compute the change in momentum.
The formula for calculating the change in momentum is given by,
\[\begin{align}
  & \Delta p=m{{V}_{f}}-m{{V}_{i}} \\
 & \Delta p=m({{V}_{f}}-{{V}_{i}}) \\
\end{align}\]
Substitute the obtained values in the above equation.
\[\begin{align}
  & \overset{\to }{\mathop{\Delta p}}\,=m({{V}_{f}}-{{V}_{i}}) \\
 & \overset{\to }{\mathop{\Delta p}}\,=0.2(-4i-6j) \\
 & \overset{\to }{\mathop{\Delta p}}\,=-0.8i-1.2j \\
\end{align}\]
We have obtained the vector form of the change in momentum value.
Now we need to find the magnitude of the change in momentum
The magnitude of the change in momentum is given by
\[\begin{align}
  & |\overset{\to }{\mathop{\Delta p}}\,|=\sqrt{{{(-0.8)}^{2}}+{{(-1.2)}^{2}}} \\
 & |\overset{\to }{\mathop{\Delta p}}\,|=\sqrt{0.64+1.44} \\
 & |\overset{\to }{\mathop{\Delta p}}\,|=\sqrt{2.08} \\
 & |\overset{\to }{\mathop{\Delta p}}\,|=1.44 \\
\end{align}\]
As the value of the change in the momentum is equal to 1.44 .

Thus, the option (C) is correct.

Note:
The things to be on your finger-tips for further information on solving these types of problems are: In the case of vector form questions, in order to find the magnitude of unknown terms we need to perform the last operation as above. Here also, we need to take care of the units given.