Drag and air resistance Transitional Reasoning Practice

Questions

1. 1.

   An engineering team evaluates a streamlined prototype vehicle that accelerates from rest along a horizontal test track under a constant engine thrust force $F$. The vehicle experiences an opposing aerodynamic drag force modeled by $F_D = C v^2$, where $C$ is a constant aerodynamic parameter and $v$ is the speed, with rolling resistance being negligible. To double the maximum steady speed that the prototype can maintain on the track, by what factor must the engine thrust force be increased, and what is the net horizontal force acting on the vehicle once it reaches this new maximum speed?

   • A) The thrust force must be increased by a factor of 2, and the net horizontal force is 0.
   • B) The thrust force must be increased by a factor of 4, and the net horizontal force equals the new thrust force.
   • C) The thrust force must be increased by a factor of 2, and the net horizontal force equals the original thrust force.
   • D) The thrust force must be increased by a factor of 4, and the net horizontal force is 0.
2. 2.

   In a physics laboratory, students drop a lightweight sphere of mass $m$ from rest in still air, where the opposing drag force is modeled by $F_D = k v^2$, with $k$ being a constant and $v$ the downward speed. Using a motion sensor, they record the sphere's downward acceleration $a$ and downward speed $v$ throughout the fall, and then plot $a$ on the vertical axis against $v^2$ on the horizontal axis. Based on Newton's laws of motion, which statement correctly identifies the physical quantities represented by the vertical intercept (where $v^2 = 0$) and the horizontal intercept (where $a = 0$) of this linear graph?

   • A) The vertical intercept equals zero, and the horizontal intercept equals the terminal speed, $v_t$.
   • B) The vertical intercept equals the gravitational acceleration $g$, and the horizontal intercept equals the square of the terminal speed, $v_t^2$.
   • C) The vertical intercept equals the gravitational acceleration $g$, and the horizontal intercept equals the terminal speed, $v_t$.
   • D) The vertical intercept equals the square of the terminal speed, $v_t^2$, and the horizontal intercept equals the gravitational acceleration $g$.
3. 3.

   Two aerodynamic test spheres, P and Q, have identical external dimensions and surface textures, but sphere Q has four times the mass of sphere P. Both spheres are dropped from rest from a high altitude in still air, experiencing a drag force modeled by $F_D = C v^2$, where $C$ is identical for both spheres. Once each sphere attains its constant terminal speed, how does the terminal speed of sphere Q compare to that of sphere P, and what is the net force acting on each sphere?

   • A) The terminal speed of Q is 4 times that of P, and the net force on Q is 4 times the net force on P.
   • B) The terminal speed of Q is 2 times that of P, and the net force on Q is 4 times the net force on P.
   • C) The terminal speed of Q is 4 times that of P, and the net force on each sphere is zero.
   • D) The terminal speed of Q is 2 times that of P, and the net force on each sphere is zero.
4. 4.

   A sensor-equipped test sphere is projected vertically upward into the air where air resistance opposes its motion. At a specific instant during its upward flight, the drag force acting on the sphere equals $0.40$ times its weight. According to Newton's laws of motion, what is the acceleration of the sphere at this instant, and what is its acceleration at the exact moment it momentarily comes to rest at its maximum height?

   • A) $0.60g$ downward during the upward flight, and $0$ at maximum height
   • B) $1.40g$ downward during the upward flight, and $0$ at maximum height
   • C) $1.40g$ downward during the upward flight, and $1.00g$ downward at maximum height
   • D) $0.60g$ downward during the upward flight, and $1.00g$ downward at maximum height
5. 5.

   A skydiver of total mass $m$ falls vertically in a stable belly-to-earth position and reaches a steady terminal speed. At this moment, the skydiver deploys a parachute, which immediately increases the upward drag force to three times the skydiver's weight ($3mg$). According to Newton's laws of motion, what is the magnitude and direction of the skydiver's acceleration immediately after the parachute opens, and how does the downward speed change right after this instant?

   • A) Magnitude $3g$ directed upward, and the downward speed decreases
   • B) Magnitude $2g$ directed downward, and the downward speed increases
   • C) Magnitude $2g$ directed upward, and the downward speed decreases
   • D) Magnitude $g$ directed downward, and the downward speed decreases
6. 6.

   A scientific drop capsule is released from rest in a high-altitude test facility where air resistance is modeled by the quadratic drag relationship $F_d = D v^2$, with $D$ being a constant and $v$ the downward speed. As the capsule accelerates downward, it approaches a terminal velocity $v_t$. At the instant when the capsule reaches a speed equal to $0.50 v_t$, what is the magnitude of its downward acceleration in terms of the gravitational acceleration $g$?

   • A) $0.25 g$
   • B) $0.50 g$
   • C) $0.67 g$
   • D) $0.75 g$

Answers and step-by-step solutions appear after you finish the test.