A model spaceship is sacrificed through freefall from 80,000 m above an asteroid and attains a velocity of 54 m/s before crashing.

 

Determine the thrust required to land safely on the asteroid if your 1,756 kg ship freefalls for 31,662 m and then thrusters create a constant force during the 47,818 m remaining distance to arrive at the surface with a velocity between 0 and 2 m/s.

To solve this, we will break the problem down into stages:

1. **Stage 1: Freefall**
   - During this stage, the spaceship freefalls 31,662 m from rest. 
   - We need to calculate the velocity at the end of this freefall.

2. **Stage 2: Thrust-Assisted Descent**
   - The spaceship uses its thrusters to decelerate over the remaining 47,818 m.
   - We need to calculate the required thrust that will slow the spaceship to a velocity between 0 and 2 m/s by the time it reaches the surface.

---

### Stage 1: Freefall

We can use the following equation for velocity under freefall:

\[
v^2 = u^2 + 2as
\]

where:
- \( v \) is the final velocity at the end of freefall,
- \( u = 0 \, \text{m/s} \) is the initial velocity,
- \( a \) is the acceleration due to gravity (assumed constant for this asteroid),
- \( s = 31,662 \, \text{m} \) is the distance fallen.

The final velocity \(v\) after freefall will serve as the initial velocity for Stage 2.

---

### Stage 2: Thrust-Assisted Descent

In this stage, we need to apply a constant thrust force to decelerate the spaceship over the remaining distance of 47,818 m, to reduce the velocity to between 0 and 2 m/s. We can use the kinematic equation:

\[
v_f^2 = v_i^2 + 2ad
\]

where:
- \( v_f \) is the final velocity (between 0 and 2 m/s),
- \( v_i \) is the velocity after the freefall from Stage 1,
- \( a \) is the required deceleration (due to the thrust),
- \( d = 47,818 \, \text{m} \) is the distance over which the deceleration occurs.

Once we find \(a\), we can calculate the required thrust \( F \) using Newton's second law:

\[
F = m(a + g)
\]

where \( m = 1,756 \, \text{kg} \) is the mass of the spaceship and \( g \) is the gravitational acceleration on the asteroid.

---

### Steps:
1. Calculate the velocity after freefall in Stage 1.
2. Use this velocity to determine the required deceleration over the remaining distance.
3. Calculate the required thrust to achieve this deceleration.

I'll calculate this now.### Calculations:

1. **Velocity after freefall (Stage 1)**: 
   - After freefalling for 31,662 meters, the spaceship attains a velocity of approximately **788.17 m/s**.

2. **Deceleration needed (Stage 2)**: 
   - To reduce the velocity from 54 m/s to a maximum of 2 m/s over the remaining 47,818 meters, the required deceleration is approximately **0.0304 m/s²**.

3. **Required thrust**: 
   - The thrust required to achieve this deceleration and land safely is approximately **17,280 N** (Newtons).

Would you like more details or have any questions about this? 

### Related Questions:
1. What factors could affect the gravity on an asteroid compared to Earth?
2. How would the required thrust change if the gravity were different?
3. How could atmospheric conditions impact a spaceship's descent on a planet?
4. What happens if the spaceship's thrusters cannot achieve the necessary deceleration?
5. How do we calculate safe landing velocities for different planetary bodies?

#### Tip:
Always ensure the gravitational constant \(g\) is adjusted for specific celestial bodies to get accurate results in space travel calculations.

A model spaceship is sacrificed through freefall from 80,000 m above an asteroid and attains a velocity of 54 m/s before crashing. Determine the thrust required to land safely on the asteroid if your 1,756 kg ship freefalls for 31,662 m and then thrusters create a constant force during the 47,818 m remaining distance to arrive at the surface with a velocity between 0 and 2 m/s.

Calculate the thrust required to land a spaceship safely on an asteroid after freefalling from 80,000 m. Using kinematic equations and Newton's Second Law, this solution explains how to compute deceleration and the necessary thrust force for a 1,756 kg ship to land at a velocity between 0 and 2 m/s. Suitable for high school students in Grades 10-12.

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