Physics

System of Particles and Rotational Motion

Conservation of linear momentum

A shell flying with velocity v=500 {~m} / {s} bursts into thre

Question

Hard

Solving time: 4 mins

A shell flying with velocity $v = 500 m / s$ bursts into three identical fragments so that kinetic energy of the system increases $1.5$ times. What maximum velocity can one of the fragments obtain ?

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Uploaded: 2023-08-19T13:19:06.000Z
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Description: A shell flying with velocity v=500 m/s bursts into three identical fragments so that kinetic energy of the system increases 1.5 times. What maximum velocity can one of the fragments obtain ?

Views: 5,637 students

Updated on: Aug 19, 2023

Book:

Physics Galaxy Mechanics Vol 1 (Ashish Arora)

Text solutionVerified

3 v = v_{1} + v_{2} + v_{3}

where

v_{1}, v_{2}

and

v_{3}

are the velocities of its fragments. From the energy conservation

3 η v^{2} = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Now

\tilde{v_{i}} or v_{i C} = v_{i} - v_{C} = v_{i} - v \tilde{v} + \tilde{v_{2}} + v_{3} = 0

Using (3) and (4) in (2), we get

3 η v^{2} = (v + \tilde{v_{1}})^{2} + (v + \tilde{v_{2}})^{2} + (v - \tilde{v_{1}} - \tilde{v_{2}})^{2} = 3 v^{2} + 2 \tilde{v}_{1}^{2} + 2 \tilde{v}_{2}^{2} + 2 \tilde{v_{1}} \cdot \tilde{v_{2}}

or,

2 \tilde{v}_{1}^{2} + 2 \tilde{v}_{1} \tilde{v}_{2} cos θ + 2 \tilde{v}_{2}^{2} + 3 (1 - η) v^{2} = 0

If we have had used

\tilde{v} = - \tilde{v_{1}} - \tilde{v}

, then Eq. 5 were contain

\tilde{v}_{3}

instead of

\tilde{v}_{2}

and so on. For

\tilde{v}_{1}

to be real in Eq. (5)

4 \tilde{v}_{2}^{2} cos^{2} θ \geq 8 (2 \tilde{v}_{2}^{2} + 3 (1 - η) v^{2}) or 6 (η - 1) v^{2} \geq (4 - cos^{2} θ) \tilde{v}_{2}^{2}

So,

\tilde{v}_{2} \leq v \frac{6 ( η - 1 )}{4 - c o s ^{2} θ} or \tilde{v}_{2 (m a x)} - 2 (η - 1) v

Hence

v_{2 (m a x)} = ∣ v + \tilde{v} ∣_{m a x} = v + 2 (η - 1) v

- v (1 + 2 (η - 1)) = 1 km / s

Thus owing to the symmetry

v_{1 (m a x)} = v_{2 (m a x)} = v_{3 (m a x)} = v (1 + 2 (η - 1)) = 1 km / s

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A shell flying with velocity

v = 500 m / s

bursts into three identical fragments so that kinetic energy of the system increases

1.5

times. What maximum velocity can one of the fragments obtain ?

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Question Text	A shell flying with velocity $v = 500 m / s$ bursts into three identical fragments so that kinetic energy of the system increases $1.5$ times. What maximum velocity can one of the fragments obtain ?
Updated On	Aug 19, 2023
Topic	System of Particles and Rotational Motion
Subject	Physics
Class	Class 11
Answer Type	Text solution:1 Video solution: 1
Upvotes	213
Avg. Video Duration	3 min

A shell flying with velocity $v = 500 m / s$ bursts into three identical fragments so that kinetic energy of the system increases $1.5$ times. What maximum velocity can one of the fragments obtain ?

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A shell flying with velocity v=500 m/s bursts into three identical fragments so that kinetic energy of the system increases 1.5 times. What maximum velocity can one of the fragments obtain ?

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A shell flying with velocity $v = 500 m / s$ bursts into three identical fragments so that kinetic energy of the system increases $1.5$ times. What maximum velocity can one of the fragments obtain ?