Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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16a0002     Projectile motion with air resistance     16a0002
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The force of air on a moving object can be quite complicated. Imagine the force of air on a spinning golf ball, for example, which can be hooked, sliced, hit with top-spin or back-spin, etc. However, we are going to use a model for air resistance that assumes that the force of air on a certain ball is

where the coefficient is in units of and speed is in m/s. Note that for low speeds, this force is very small in comparison to the gravitational force on a more massive object. However, for a 0.002-kg nerf ball thrown with an initial speed of 5.55 m/s, the force of air on the ball at the instant it is thrown is 0.0308 N. That is bigger than its weight! Therefore, the force of air on the ball cannot be neglected.

  1. Suppose you toss a 0.002 kg nerf ball to a friend. The ball leaves your hand with a speed of 5.55 m/s at an angle, with respect to the horizontal, of . Define the origin to be the position of the ball at the instant it leaves your hand. Calculate the position and velocity of the ball between t = 0 and t = 0.1 s:

  2. If the ball leaves your hand at a height of 1.5 m from the ground, what is the clock reading when the ball hits the ground and how far from your hand (horizontally) does the ball land?

 

16a0001     Force by air resistance on a skydiver     16a0001
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Suppose that a skydiver falls from an airplane, with her arms and legs outward in a horizontal plane as she falls downward. Her mass is 65 kg and she eventually reaches terminal speed of 55 m/s. Assume that the drag force on her has a magnitude where is a constant that depends on the density of air, the cross-sectional area of the skydiver, and the drag coefficient of the skydiver.

  1. What is the constant for the skydiver?
  2. Suppose that the constant D is approximately the same for another skydiver of mass . What will be his terminal speed?

 

1610001     Stiffness of an airbag on a Mars rover     1610001
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When the Mars rovers Spirit and Opportunity landed on Mars, they landed via an airbag. Basically, as they were falling, an airbag deployed and they bounced off the surface of Mars until they finally came to rest. So that they didn't have too much kinetic energy upon impact, thrusters were initially fired in order to slow them down before the airbags were deployed.

The airbag can be modeled as a spring because when the rover and airbag hit the surface of mars, the airbag compresses much like a spring would compress.

Suppose that a 250 kg rover is dropped onto the surface of Mars. At a height of 10 m, the thrusters are turned off and it has a downward speed of 5 m/s. When it hits the surface of Mars, the airbag compresses 0.5 m at full impact before the system then rebounds upward. The acceleration due to gravity near the surface of Mars is . What is the "stiffness" of the airbag if you model the airbag as an ideal spring?

 

1610002     An oscillating cart on a track.     1610002
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A spring is attached between the end of a low-friction track and the front of a 1.2 kg cart. A second identical spring is attached to the back of the cart and the other end of the track. When in equilibrium, each spring is stretched 0.4 m from its unstretched position. The cart is displaced 0.15 m from equilibrium and released from rest. It oscillates with a period of 3.0 s. x is the position of the cart at any time t, with x = 0 defined as the equilibrium position of the cart.

  1. What is the angular frequency of the system?
  2. What is the effective spring stiffness of the system?
  3. What is the total energy of the oscillator?
  4. What is the maximum speed of the cart?
  5. When the cart is 0.075 m from its equilibrium position, what percentage of the total energy is elastic potential energy and what percentage is kinetic energy?

 

1640001     Equilibrium temperature of a cup of coffee     1640001
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If 0.5 kg of coffee at is poured into a 1-kg glass mug ( ) at , what will be the equilibrium temperature of the system if the system is thermally insulated from its surroundings?

 

1920001     A "rear-end" collision of two carts on a track.     1920001
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In the lab, a 1.0 kg cart traveling in the +x direction with a speed of 0.5 m/s collides with a 2.0 kg cart that is moving in the +x direction at 0.2 m/s. If the velocity of the more massive cart after the collision is 0.3 m/s, what is the change in internal energy of the system of the two carts?

 

1820001     Elastic collision of an alpha particle and an oxygen nucleus.     1820001
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A car of mass 500 kg and a truck of mass 1200 kg are moving directly toward each other at the same speed of 20 m/s. Define the +x direction to the direction of the car.

  1. What is the velocity of the center of mass of the system, defined to be the car and truck?
  2. At the moment when the vehicles are 100 m apart, what is the location of the center of mass of the system? Treat the vehicles as point particles.

 

1190004     Change in momentum of a rebounding tennis ball     1190004
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A 57-gram tennis ball with a velocity of m/s rebounds from a wall with a velocity of m/s. What is the tennis ball's change in momentum in kg m/s?

 

1190001     Change in momentum of a golf ball (as it rims out)     1190001
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A 0.045-kg golf ball is rolling toward the hole with a velocity m/s. A s, it hits the edge of the hole; it then travels around the hole ("rimming out") and continues rolling. It leaves the hole at s with a velocity m/s. A picture of the ball at 0.1 s intervals is shown in the figure below.


Figure: The path of a golf ball as it travels around the rim of the hole on a putting green.

(View a simulation at

http://linus.highpoint.edu/~atitus/mandi/physlets/1190001/1190001.html

to visualize the motion.)

  1. Sketch the golf ball's initial momentum and final momentum vectors for the time interval that the ball and hole interact.
  2. Sketch the golf ball's change in momentum during the time interval that it interacts with the hole.
  3. What is the change in momentum of the golf ball as a result of interacting with the hole?
  4. What is the magnitude and direction of the change in momentum of the golf ball?

 

1190005     Change in momentum of the Huygens spaceprobe upon impact with Titan     1190005
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The following quote is from http://spaceflightnow.com/cassini/050115science.html

Huygens entered Titan's thick nitrogen atmosphere around 5:13 a.m. Friday. John Zarnecki, principal investigator of the surface science package, said it took the spacecraft two hours 27 minutes and 50 seconds to complete its parachute descent to the surface. It hit that surface at a velocity of 10.1 mph and experienced a very brief impact deceleration of 15 Gs. The jolt knocked one sensor off line, but it came back to life on its own a few minutes later.

A "penetrometer" on the bottom of the probe extended six inches into the frigid soil. That data, coupled with the deceleration experienced by Huygens as it hit the ground, provided new insights into the nature of the surface material at the landing site.

"What we're seeing is, we think, a material which might have a thin crust followed by a region of relatively uniform consistency," said Zarnecki. "In terms of this (impact) force, the closest analog that I can give you - and remember, this is not suggesting these are the materials we hit, but that the mechanical consistency is similar - then I would say wet sand or clay are materials which give a similar sort of trace."

The mass of the Huygens spacecraft is 192 kg.

  1. What was the momentum of the Huygens just before impact with the surface of Titan?
  2. What was Huygens' change in momentum as a result of the impact?

 


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