Calculus in physics (derivatives and integrals) Transitional Reasoning Practice

Questions

1. 1.

   A motorized positioning stage in an optical laboratory moves along a linear track, with its position coordinate described by $x(t) = 2t^3 - 9t^2 + 12t + 5$, where x is in meters and t is in seconds for $t \ge 0$. The system control unit applies brakes when the stage momentarily comes to rest at its second stopping point. Using derivative calculus, what is the acceleration of the stage at this precise instant?

   • A) -6 m/s2
   • B) 0 m/s2
   • C) 18 m/s2
   • D) 6 m/s2
2. 2.

   A sensor-equipped drone moves along a straight testing corridor with a time-dependent velocity given by $v(t) = 12t - 3t^2$, where $v$ is in meters per second and $t$ is in seconds for $t \ge 0$. Using calculus principles, what is the total displacement of the drone from $t = 0\text{ s}$ up to the instant it attains its maximum forward velocity?

   • A) 12 m
   • B) 16 m
   • C) 24 m
   • D) 32 m
3. 3.

   An experimental cart moves along a straight calibration track with a time-dependent acceleration given by $a(t) = 12t - 6$, where $a$ is in meters per second squared and $t$ is in seconds. At $t = 0\text{ s}$, the cart passes a sensor with an initial velocity of $5\text{ m/s}$. Based on integral calculus principles, what is the net displacement of the cart from $t = 0\text{ s}$ to $t = 2\text{ s}$?

   • A) 4 m
   • B) 10 m
   • C) 14 m
   • D) 17 m

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