For a safe re-entry into the Earth's atmosphere the pilots of a space capsule must reduce their speed from 2.6 104 m/s to 1.0 104 m/s. The rocket engine produces a backward force on the capsule of 1.7 105 N. The mass of the capsule is 3800 kg. For how long must they fire their engine? [Hint: Ignore the change in mass of the capsule due to the expulsion of exhaust gases.]

To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

F = m * a

In this case, the net force is the backward force produced by the rocket engine, and we want to find the time (t) for which this force needs to be applied. We also know the mass (m) of the capsule.

First, let's find the acceleration (a) experienced by the capsule. Rearranging the equation, we have:

a = F / m

Substituting the given values, we get:

a = (1.7 * 10^5 N) / (3800 kg)

Calculating this, we find:

a ≈ 44.74 m/s^2

Now, we can use the equation of motion to find the time needed to change the velocity. The equation of motion for constant acceleration is:

v = u + a * t

Where:
- v is the final velocity (1.0 * 10^4 m/s)
- u is the initial velocity (2.6 * 10^4 m/s)
- a is the acceleration (44.74 m/s^2)
- t is the time we want to find

Rearranging the equation, we have:

t = (v - u) / a

Substituting the given values, we get:

t = (1.0 * 10^4 m/s - 2.6 * 10^4 m/s) / 44.74 m/s^2

Evaluating this expression, we find:

t ≈ -44.55 s

Since time cannot be negative, the negative sign indicates that the capsule is decelerating. Therefore, we take the magnitude of the time, giving us:

t ≈ 44.55 s

Therefore, the pilots must fire their engine for approximately 44.55 seconds to achieve the desired change in velocity for a safe re-entry into the Earth's atmosphere.