To determine the change in kinetic energy of the unmanned space probe after the thrust is applied, we can follow a systematic approach. First, we need to calculate the initial kinetic energy, then find the final velocity after the thrust is applied, and finally compute the final kinetic energy. The change in kinetic energy will be the difference between the final and initial kinetic energies.
Step 1: Calculate Initial Kinetic Energy
The formula for kinetic energy (KE) is given by:
KE = 0.5 * m * v²
Where:
- m = mass of the space probe (1780 kg)
- v = initial velocity (349 m/s)
Plugging in the values:
KE_initial = 0.5 * 1780 kg * (349 m/s)²
Calculating this gives:
KE_initial = 0.5 * 1780 * 121801
KE_initial = 108,000,000 J (approximately)
Step 2: Determine the Final Velocity
Next, we need to find the final velocity after the thrust has been applied. The thrust force (F) can be used to calculate the acceleration (a) using Newton's second law:
F = m * a
Rearranging gives:
a = F / m
Substituting the values:
a = 3940 N / 1780 kg
a ≈ 2.21 m/s²
Now, we can find the change in velocity (Δv) using the formula:
Δv = a * t
Where t is the time the thrust is applied (82.1 s):
Δv = 2.21 m/s² * 82.1 s
Δv ≈ 181.5 m/s
Now, we can find the final velocity (v_final):
v_final = v_initial + Δv
v_final = 349 m/s + 181.5 m/s
v_final ≈ 530.5 m/s
Step 3: Calculate Final Kinetic Energy
Using the final velocity, we can now calculate the final kinetic energy:
KE_final = 0.5 * m * v_final²
KE_final = 0.5 * 1780 kg * (530.5 m/s)²
KE_final = 0.5 * 1780 * 281,130.25
KE_final ≈ 250,000,000 J (approximately)
Step 4: Calculate Change in Kinetic Energy
The change in kinetic energy (ΔKE) is the difference between the final and initial kinetic energies:
ΔKE = KE_final - KE_initial
ΔKE = 250,000,000 J - 108,000,000 J
ΔKE ≈ 142,000,000 J
In summary, the change in kinetic energy of the unmanned space probe after the thrust is applied is approximately 142,000,000 joules. This significant increase in kinetic energy illustrates how even a relatively small thrust can have a substantial impact on the velocity of a spacecraft over time.
