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In the movie the martian what was the purpose of the hermes crew in blasting a part of the ship?

Was that explosion on the front or back of the ship?

After the explosion what happened to the speed of the hermes ship?

Has the radius of its orbit increased or decreased?

Got closer to the MAV?

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I do not have the book, so lets look at the script, in particular page 99 and following:

JOHANSSEN Velocity 850, altitude 1843 --
[...]
JOHANSSEN He’s well below target altitude.
LEWIS How far below?
JOHANSSEN Working on it
[...]
JOHANSSEN I have interval pings. Intercept velocity will be 11 meters per second...
BECK (OVER COMMS) I can make that work.
JOHANSSEN Distance at intercept will be -- (goes pale) We’ll be 68 kilometers apart.

The MAV underperformed. They had stripped it down, removed weight, but with the chute not holding etc. They did not get enough performance out of the MAV to be high enough.

LEWIS (OVER COMMS) We’re working on getting you. There was a complication during launch.
MARK Yeah. The canvas didn’t hold... I think it ripped early in the ascent.
LEWIS That’s consistent with what we saw.
MARK (OVER COMMS) How bad is it, Commander?
LEWIS We’ve corrected the intercept range, but we’ve got a problem with the intercept velocity.
MARK (OVER COMMS) How big a problem?
LEWIS 42 meters per second.

Mark Watney suggest punching a hole in his suit to gai thrust:

MARK I could find something sharp in here and poke a hole in the glove of my EVA suit. I could use the escaping air as a thruster and fly my way to you. Since the source is on my arm, I could direct it pretty easy.

They discuss this idea (briefly) before coming to the conclusion:

LEWIS Maybe it’s not the worst idea.
MARTINEZ No, it actually is. The worst idea. Ever.
LEWIS Not his part. But using atmosphere as thrust...

So, to answer the first part of your question:

In the movie the martian what was the purpose of the hermes crew in blasting a part of the ship?

Releasing the atmosphere creates thrust. It doesn't matter if you push exhaust through a nozzle or atmosphere through a tubular ship segment - both create forward thrust:

LEWIS I need to know what happens if we blow the VAL.
Both Martinez and Johanssen straighten up. What?
MARTINEZ You want to open the Vehicular Airlock?
LEWIS It would give us a good kick.
MARTINEZ Yeah. And it might blow the nose of the ship off in the process.
JOHANSSEN And... all the air would leave. And we need air. To not die.

Now, the next three of your question are answered at once:

Was that explosion on the front or back of the ship?
After the explosion what happened to the speed of the hermes ship?
Has the radius of its orbit increased or decreased?

The crew discusses this shortly after:

MARTINEZ But we’d still have the same problem as Watney. We can’t direct the thrust.
LEWIS We don’t have to. The VAL is in the nose. We just point the ship at Mark.
MARTINEZ (reading the numbers) A breach at the VAL would decelerate us 29 meters per second.
JOHANSSEN Which gives us a relative velocity of 13 meters per second.

Mark was slower and thus in a lower orbit. The blew the airlock in the nose - the front - of the ship, and thus decelerated. They were not in orbit, but on an hyperbolic escape path around mars - they did not slow down enough to get into orbit. But this hyperbolic path was indeed lowered to get closer to the MAV.

The result of the operation is described later:

JOHANSSEN (OVER COMMS) 12 meters per second.
LEWIS Copy.
[...]
LEWIS (OVER COMMS) What’s our intercept range?
JOHANSSEN 312 meters.

Now, for your last question:

Got closer to the MAV?

If you look at two orbits - or in this case, hyperbolic paths - then you recognize this question doesn't really make sense as asked. They successfully managed to lower the intercept distance. At the beginning, the closest approach they had to Watney when their paths crossed was 68km. After blowing the nose and thus adjusting their course, the closest approach - the intercept range - came down to 312 meters.

If you have access to it, the Mars launch sequence in the movie starts at about 1:56:00, its pretty well explained in-movie.

As an aside, this might be better suited to be asked at Movies & TV Stack Exchange or Science Fiction & Fantasy Stack Exchange.

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  • $\begingroup$ R=Gm/v^2 Therefor if they decelarate the radius will increase. In the movie when they try to catch Wayne the ship is getting further way. $\endgroup$ – tomas figueiredo Mar 20 at 12:52
  • $\begingroup$ @tomasfigueiredo Unfortunately, that is not how orbital mechanics work. The equation you have given describes orbital velocity for a circular orbit. In a perfectly circular orbit, if you decelerate (thrust prograde), you lower the opposite side of the orbit, which will be the periapsis, and the apoapsis will be at the point where the thrust was applied. Your speed at periapsis will be higher then the speed at apoapsis, and the orbit will be eccentric, giving an higher average orbital velocity. In case of a hyperbola, decelerating likewise also lowers eccentricity and the periapsis. $\endgroup$ – Polygnome Mar 20 at 14:25
  • $\begingroup$ @tomasfigueiredo Also, if you read the quotes from the script I have provided, they start with 68 kilometres separation at closest approach (intercept range), and their deceleration maneuver lowers this to 312 metres. this is due to marks apoapsis being far lower then intended, and them lowering their periapsis towards his apoapsis. $\endgroup$ – Polygnome Mar 20 at 14:26
  • $\begingroup$ Minor point: Watney's craft has no apoapsis, it's on a hyperbolic escape trajectory. $\endgroup$ – notovny Mar 23 at 13:03

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