Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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(N)=# of solutions
1260002     Forces on a Moon lander as it is landing.     1260002
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Question

 

A 500-kg Moon lander has a velocity of 10 m/s downward when it is a certain height above the ground. 1 minute later, it is 1 m from the ground and its speed is 0.5 m/s downward. Assume that the thruster on the lander fires upward with a constant force. Moon's gravitational field strength near its surface is 1.6 N/kg.

  1. What is the force by the exhaust on the lander (i.e. thrust)?
  2. What was its initial height, when its speed was 5 m/s downward?

 



Solution

 

(a) Begin by drawing a picture, showing the initial momentum, final momentum, and the change in momentum of the lander.


Figure: A Moon lander slows down with a constant net force.


Figure: The change in momentum of the lander.

Apply the momentum principle. Here are the steps:

(1) define the system:

(2) apply the momentum principle to the system. The change in momentum and the time interval can be used to calculate the net force.

Substitute the given quantities

Check your answer. Note that the net force points upward, in the +y direction. This is consistent with the direction of the change in momentum from our picture. Though the lander is falling, it is slowing down. This indicates a net force that is upward.

To calculate an individual force acting on the system, you must apply the definition of net force as the vector sum of all forces acting on the system. Here are the steps:

(1) List the objects in the surroundings that exert forces on the system. This list includes any objects that makes contact with the system and any objects that exerts "force at a distance" such as a gravitational force or an electrostatic force or a magnetic force, for example. Here are the objects in the surroundings that exert forces on the system.

(2) Sketch a force diagram showing the forces acting on the system with approximate magnitudes and directions if they are known.


Figure: The force diagram showing the forces on the lander.

Note that the thrust must be in the direction shown so that when adding the vectors tail to head, the resultant (or net force) is upward, in the same direction as the change in momentum, in accordance with the momentum principle. The sum of the forces is shown below.


Figure: The sum of the forces on the lander give the correct net force.

(3) Apply the definition of net force and solve for the unknown force. Sometimes you have to do this in component form and sometimes you can do it in vector form. In this case, we'll use vector form, though only the y-component is needed since there is no motion or forces in the x and z directions.

The thrust on the lander is upward and equal to 879 N. It's larger than the gravitational force on the lander (800 N) which gives an upward net force, in agreement with the free-body diagram. Since the lander is falling during the time interval and since the net force is upward, the lander slows down.

(b) To determine the initial height of the lander, apply the definition of average velocity.

For the average velocity, use the arithmetic mean of the initial and final velocity of the lander.

Substitute the average velocity and solve for the initial position of the lander.

It's certainly higher than 1 m, as expected since the lander is falling toward the surface.