A lunar lander is making its descent to Moon Base I (Fig. E2.402.40). The lander descends slowly under the retro-thrust of its descent engine. The engine is cut off when the lander is 5.05.0 m above the surface and has a downward speed of 0.80.8 m/s. With the engine off, the lander is in free fall. What is the speed of the lander just before it touches the surface? The acceleration due to gravity on the moon is 1.61.6 m/s.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 40

Chapter 2, Problem 40

A lunar lander is making its descent to Moon Base I (Fig. E $2.40$ ). The lander descends slowly under the retro-thrust of its descent engine. The engine is cut off when the lander is $5.0$ m above the surface and has a downward speed of $0.8$ m/s. With the engine off, the lander is in free fall. What is the speed of the lander just before it touches the surface? The acceleration due to gravity on the moon is $1.6$ m/s.

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Identify the initial conditions: The lander is 5.0 m above the surface with a downward speed of 0.8 m/s. The acceleration due to gravity on the moon is 1.6 m/s².

Use the kinematic equation for velocity in free fall: v² = u² + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance fallen.

Substitute the known values into the equation: u = 0.8 m/s, a = 1.6 m/s², and s = 5.0 m.

Calculate the expression inside the square root: u² + 2as = (0.8 m/s)² + 2 * (1.6 m/s²) * (5.0 m).

Solve for v by taking the square root of the result from the previous step: v = √[(0.8 m/s)² + 2 * (1.6 m/s²) * (5.0 m)].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Free Fall

Free fall occurs when an object is falling solely under the influence of gravity, with no other forces acting on it. In this scenario, once the lunar lander's engine is cut off, it enters free fall, meaning it will accelerate downward at the acceleration due to gravity on the Moon, which is 1.6 m/s². This concept is crucial for determining the lander's final speed just before it touches the surface.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this case, we can use the equation that relates initial velocity, final velocity, acceleration, and distance: v² = u² + 2as, where 'v' is the final velocity, 'u' is the initial velocity, 'a' is the acceleration, and 's' is the distance fallen. This equation will help calculate the lander's speed just before impact.

Initial Velocity

The initial velocity is the speed of an object at the start of a time interval. For the lunar lander, its initial velocity when the engine is cut off is 0.8 m/s downward. This value is essential for calculating the final speed just before the lander touches the surface, as it influences how fast the lander will be moving after falling the remaining distance of 5.0 m.

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