Juggling with Danger

Classical Mechanics Level 4

Danny the Dangerous is juggling with bombs. He throws a bomb vertically upwards, and when it reaches its highest point, it explodes into three pieces. If all three pieces have the same speed just after the explosion, then which of the following statements are necessarily true?

A : The three pieces have equal masses.
B : The three pieces reach the ground with the same speed.
C : The three pieces reach the ground at the same time.

Note:
- Ignore air resistance.
- Do not try this stunt at home, school or anywhere else.

Image Credits: Pixabay and Wikimedia Commons Nevit Dilmen

A, C only A, B, and C None of the statements are necessarily true C only B, C only B only A only A, B only

1 solution

Pranshu Gaba
Apr 20, 2016

Relevant wiki: Conservation of Momentum

When the bomb reaches its highest point, its instantaneous velocity becomes zero, so its instantaneous momentum is also zero.

Since the explosion occurs very quickly, the total momentum of the bomb just before the explosion (which is zero) is equal to the total momentum of the bomb just after the explosion. Therefore the total momentum of the bomb after the explosion is also zero.

Let the momenta of the three pieces just after the explosion be $m_1 \vec{v_1}, \ m_2 \vec{v_2},\ m_3 \vec{v_3}$ . We know that they add up to zero.

$m_1 \vec{v_1} + m_2 \vec{v_2} + m_3 \vec{v_3} = 0$

Since the initial speeds of the three pieces are equal, the magnitudes of $\vec{v_1}, \ \vec{v_2}, \ \vec{v_3}$ are equal. Also, since the sum of three vectors is zero, they must be coplanar. Note that we do not have restrictions on the values of $m_1, \ m_2$ and $m_3$ . They do not have to be equal. Therefore Statement A may be false .

Just after the explosion, all three pieces are at the same height, say $h$ and have the same speed, say $v$ . They are all in freefall after the explosion, so they all experience acceleration due to gravity, $g$ . After falling through the same height $h$ their final speeds will be the same. Therefore Statement B is necessarily true .

It is possible for $v_1$ , $v_2$ and $v_3$ to have different vertical components. The piece which has the maximum downwards velocity will reach the ground before the other pieces. Therefore Statement C may be false . $_\square$

How do you know the direction of the initial velocities of the 3 parts after the explotion .why isnt the answer ..non of the statements are true

Chamodya Attanayake - 5 years, 1 month ago

Hi Chamodya, I do not know the direction of initial velocities of the 3 parts after the explosion. However their initial speeds are the same, and since they fall through the same height, their final speeds are also the same. Therefore only B is necessarily true.

Pranshu Gaba - 5 years, 1 month ago

Hey Pranshu Gaba, I understand your idea, but you made a considerable error. Since there is some energy released by an explosion of any kind : in this case it parts the ball in three. In this way, since we can see that there are three parts, they must have moved apart from each other. So their momentum cannot be zero. The energy before and after the reaction is very different, because the explosion "gave" its energy into kinetic energy for the three parts.

Battist Oberhößel - 5 years, 1 month ago

Hi Battist, I agree that when a bomb explodes, its stored potential energy gets converted to kinetic energy. The three pieces formed after the explosion have more kinetic energy than the bomb had before the explosion. However, the net momentum of the bomb is conserved. It is possible to have non-zero kinetic energy, and yet have zero momentum.

This is possible since momentum is a vector quantity. If two objects have moving in opposite directions with equal momenta, their momentums will be cancelled and the net momentum of both object will be zero, even though both are moving.

Suppose the pieces after explosion have masses $m$ , $m$ , and $2m$ . The $2m$ piece moves upwards, and the other two pieces move downwards. Net momentum is $(2m)v - 2(mv) = 0$ . Momentum is conserved. However the pieces have more kinetic energy than the bomb had before it exploded.

Pranshu Gaba - 5 years, 1 month ago

At the highest point, the bomb has some non zero horizontal velocity right?

A Former Brilliant Member - 5 years, 1 month ago

Since the bomb is thrown vertically upwards, the bomb will not have any horizontal velocity before the explosion. The pieces of the bomb may have non zero horizontal velocity after the explosion.

Pranshu Gaba - 5 years, 1 month ago

Nice problem. I know that you are correct that they will all reach the ground with the same speed, but I don't understand your explanation---probably it's not your intention but to me it reads like you're saying that objects which experience the same force over the same distance get the same total acceleration (when really it's the time, not the distance, that matters). Don't you need to invoke conservation of energy; they have the same kinetic + potential energy and will lose the same potential energy as they move from $h$ to $0$ , so their speed must be the same so that they have the same kinetic energy?

Also, you might want to say something in the problem that excludes the possibility of a piece having sufficient velocity to escape.

Finally, I enjoyed the graphic and the important safety warning!

Mark C - 5 years ago

Juggling with Danger

Image Credits: Pixabay and Wikimedia Commons Nevit Dilmen

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