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A bomb of 300grams at rest explodes into three equal pieces, two of them found to move perpendicular to each with 20m/s speed. Find the speed and the direction of 3rd piece.

Answer
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Hint: Calculate the angle made by the third piece of exploded bomb. Adding and squaring in the equation to obtain the speed of the third piece of the exploded bomb.

Complete step by step answer:
Let us represent the first piece of the bomb by 1 and represent the second piece by 2 and represent the third piece by 3. And represent the speed of pieces by v.
A bomb of 300grams is exploded into three equal pieces. Two of them are right angled to each. The speed of the two pieces is 20m/s.
Let us consider the figure,


The linear momentum is conserved before and after the explosion of the bomb.
So, the bomb is at rest before it explodes. Therefore, zero is the initial momentum of the bomb.
We consider that the piece 1 moves along xaxis, and the piece 2 along the yaxis, we assume that the third piece 3 moves towards the angle of θ with the negative xaxis at speed of v
The xaxis is conserving,
0=1(20)1(vcosθ)
Now we simplify the above expression.
vcosθ=20......(1)
And the yaxis is conserving,
0=1(20)1(vsinθ)
Now we simplify the above expression as,
vsinθ=20......(2)
From equation (1) and (2) we get,
tanθ=vsinθvcosθ
Now we substitute the values as,
tanθ=2020
Now, solve the above expression as,
tanθ=1

Now, solve further for angle θ.
θ=45o,
So, we calculate as,
ϕ=180o45o
Now we solve the above expression as,
ϕ=135o
The third piece moves with an angle of 135o with 1.
Now, calculate the speed of the third piece and adding equation (1) and (2),
vcosθ+vsinθ=20+20
By simplifying the above relation, we get,
vcosθ+vsinθ=40......(3)
Now we are squaring both sides to get
v2(sin2θ+cos2θ)=202+202
Now, simplify the expression.
v2(1)=400+400
Now, solve the expression further.
v2=800
We solve the value of v as,
v=202m/s

Therefore, the speed of the third piece that explodes is 202m/s.

Note: As we know that the velocity is the vector quantity and the resultant of the velocities will be calculated by using the vector addition that is it depends on the direction of the velocities.
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