The space probe Deep Space I was launched on October 24, 1998. It's mass was 474 kg. The goal of the mission was to test a new kind of engine called an ion propulsion drive. This engine generated only a weak thrust, but it could do so over long periods of time with the consumption of only small amounts of fuel. The mission was spectacularly successful. At a thrust of 52 mN how many days were required for the probe to attain a velocity of 880 m/s (1969 mi/h), assuming that the probe started from rest and that the mass remained nearly constant?
f = m * a ___ a = f / m
a = .052 N / 474 kg
v = a * t ___ t = v / a
t = 880 m/s / (.052 N / 474 kg)
t will be in sec__convert to days
To calculate the time required for the probe to attain a velocity of 880 m/s, we can use Newton's second law of motion, which states:
Force = mass × acceleration
In this case, the force exerted by the ion propulsion drive is given as 52 millinewtons (52 mN), and the mass of the probe is given as 474 kg. The acceleration can be calculated by dividing the force by the mass:
Acceleration = Force / Mass
Acceleration = 52 mN / 474 kg
Now, we can use the equation of motion to find the time required:
Velocity = Initial Velocity + (Acceleration × Time)
Given that the probe starts from rest, the initial velocity is 0 m/s. We need to solve for time, so we rearrange the equation:
Time = (Velocity - Initial Velocity) / Acceleration
Time = 880 m/s / [(52 mN / 474 kg)]
Now, let's convert the thrust force from millinewtons (mN) to newtons (N) to ensure consistent units:
52 mN = 52 × 10^(-3) N
Substituting the values:
Time = 880 m/s / [(52 × 10^(-3) N) / 474 kg]
Now, we can calculate the time:
Time = 880 m/s / (52 × 10^(-3) N / 474 kg) = (880 × 474) / 52 × 10^(-3) = 799200 / 52 × 10^(-3)
Simplifying:
Time = 15,322,054 days
Thus, it would take approximately 15,322,054 days for the probe to attain a velocity of 880 m/s, assuming the given conditions.