Let's investigate a possible vertical landing on Mars that...

Let's investigate a possible vertical landing on Mars that includes two segments: free fall followed by a parachute deployment. Assume the probe is close to the surface, so the Martian acceleration due to gravity is constant at $3.00 \mathrm{~m} / \mathrm{s}^{2}$. Suppose the lander is initially moving vertically downward at $200 \mathrm{~m} / \mathrm{s}$ at a height of $20000 \mathrm{~m}$ above the surface. Neglect air resistance during the free-fall phase. Assume it first free falls for $8000 \mathrm{~m}$. (The parachute doesn't open until the lander is $12000 \mathrm{~m}$ from the surface. See $\mathbf{r}$ Fig. $2.29 .$ ) (a) Determine the lander's speed at the end of the 8000 -m free-fall drop. (b) At $12000 \mathrm{~m}$ above the surface, the parachute deploys and the lander immediately begins to slow. If it can survive hitting the surface at speeds of up to $20.0 \mathrm{~m} / \mathrm{s}$, determine the minimum constant deceleration needed during this phase. (c) What is the total time taken to land from the original height of $20000 \mathrm{~m} ?$

Let's investigate a possible vertical landing on Mars that includes two segments: free fall followed by a parachute deployment. Assume the probe is close to the surface, so the Martian acceleration due to gravity is constant at $3.00 \mathrm{~m} / \mathrm{s}^{2}$. Suppose the lander is initially moving vertically downward at $200 \mathrm{~m} / \mathrm{s}$ at a height of $20000 \mathrm{~m}$ above the surface. Neglect air resistance during the free-fall phase. Assume it first free falls for $8000 \mathrm{~m}$. (The parachute doesn't open until the lander is $12000 \mathrm{~m}$ from the surface. See $\mathbf{r}$ Fig. $2.29 .$ ) (a) Determine the lander's speed at the end of the 8000 -m free-fall drop.
(b) At $12000 \mathrm{~m}$
above the surface, the parachute deploys and the lander immediately begins to slow. If it can survive hitting the surface at speeds of up to $20.0 \mathrm{~m} / \mathrm{s}$, determine the minimum constant deceleration needed during this phase.
(c) What is the total time taken to land from the original height of $20000 \mathrm{~m} ?$

Key Concepts

Segmented Motion Analysis

In many practical problems, the object's motion is divided into different segments, each characterized by its own acceleration (or deceleration). Analyzing each segment separately and then combining the results is crucial when changes, such as the deployment of a parachute, alter the acceleration mid-motion.

Deceleration

Deceleration refers to a reduction in the speed of an object, characterized by a negative acceleration. Determining the minimum deceleration required is critical in scenarios where an object must slow down sufficiently over a set distance to meet safety thresholds.

Uniformly Accelerated Motion

This concept involves an object moving with a constant acceleration, such as gravitational acceleration in free-fall scenarios. It is used to predict the position, velocity, and time relationships when no other forces (like air resistance) are acting significantly on the object.

Kinematic Equations

These are the mathematical formulas that connect displacement, initial velocity, acceleration, and time. They are essential for solving problems involving constant acceleration, allowing the calculation of final speed, distance traveled, and the duration of different phases of motion.

Time-of-Flight Calculation

This is the process of determining the total time taken for an object to travel over a certain distance, particularly when the motion occurs in multiple phases with different accelerations. It involves applying the kinematic equations to each phase and summing the time intervals to get the overall duration.

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