The space probe Deep Space 1 was launched on October 24, 1998. Its 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 56 mN how many days were required for the probe to attain a velocity of 725 m/s (1620 mi/h), assuming that the probe started from rest and that the mass remained nearly constant?
To calculate the time required for the space probe Deep Space 1 to attain a velocity of 725 m/s (1620 mi/h) with a thrust of 56 mN, we can use Newton's second law of motion.
The formula to calculate the time required for a change in velocity with a constant force is:
time = (change in velocity) / (acceleration)
In this case, the thrust provided by the ion propulsion drive is the force causing the acceleration of the space probe.
1. First, convert the thrust from millinewtons to newtons:
56 mN = 0.056 N
2. Then, calculate the acceleration using Newton's second law of motion:
acceleration = force / mass
Since the mass remained nearly constant:
acceleration = 0.056 N / 474 kg
3. Now, we can calculate the time required for the change in velocity:
time = (change in velocity) / (acceleration)
Given that the change in velocity is from rest (0 m/s) to 725 m/s:
time = (725 m/s - 0 m/s) / (0.056 N / 474 kg)
Simplifying the expression:
time = 725 m/s / (0.056 N / 474 kg)
4. Calculate the time in seconds:
time = 725 m/s / (0.056 N / 474 kg)
time ≈ 9.6 × 10^5 s
5. Convert the time to days:
time = 9.6 × 10^5 s / (60 s * 60 min * 24 h)
time ≈ 11.11 days
Therefore, it takes approximately 11.11 days for the Deep Space 1 probe to attain a velocity of 725 m/s with a thrust of 56 mN.