Kinematics with calculus Transitional Reasoning Practice

Questions

1. 1.

   A high-speed test carriage moves along a straight track with its position modeled by $x(t) = 2t^3 - 15t^2 + 24t + 50$, where $x$ is in meters and $t \ge 0$ is in seconds. Control engineers analyze the trial run to determine the time intervals during which the carriage's speed is increasing. During which time intervals is the carriage speeding up?

   • A) $0 < t < 1\text{ s}$ and $t > 4\text{ s}$
   • B) $1 < t < 4\text{ s}$
   • C) $t > 2.5\text{ s}$
   • D) $1 < t < 2.5\text{ s}$ and $t > 4\text{ s}$
2. 2.

   An automated vertical research probe is launched upward from ground level ($y = 0\text{ m}$) at $t = 0\text{ s}$ along a straight vertical test track. Telemetry systems record its vertical velocity as $v(t) = 20t - 6t^2$, where $v$ is in meters per second and $t$ is in seconds. What is the acceleration of the probe at the exact instant it returns to ground level?

   • A) $-20\text{ m/s}^2$
   • B) $-40\text{ m/s}^2$
   • C) $-50\text{ m/s}^2$
   • D) $+40\text{ m/s}^2$
3. 3.

   A robotic shuttle moves along a straight automated assembly track with its position modeled by $x(t) = \frac{1}{4}t^4 - 2t^3 + 4t^2$, where $x$ is in meters and $t$ is in seconds for $t \ge 0$. Control engineers monitor the shuttle as it slows down and changes direction during its cycle. What is the displacement of the shuttle from the first moment it momentarily comes to rest after $t = 0\text{ s}$ to the second moment it comes to rest?

   • A) $+4\text{ m}$
   • B) $-4\text{ m}$
   • C) $-8\text{ m}$
   • D) $0\text{ m}$
4. 4.

   An autonomous transport cart moves along a straight aisle with a velocity modeled by $v(t) = 12t - 3t^2\text{ m/s}$ for $t \ge 0$, where $t$ is in seconds. At time $t = 0\text{ s}$, the cart is located at an initial position of $x(0) = 5\text{ m}$. What is the position of the cart along the aisle at the instant its acceleration reaches zero?

   • A) 12 m
   • B) 16 m
   • C) 21 m
   • D) 37 m
5. 5.

   An experimental rocket sled travels along a straight test track with its position given by $x(t) = t^3 - 6t^2 + 24t$, where $x$ is in meters and $t$ is in seconds for $t \ge 0$. Telemetry engineers analyze the motion over the initial interval from $t = 0\text{ s}$ to $t = 6\text{ s}$. What is the instantaneous acceleration of the sled at the non-zero time when its instantaneous velocity equals its average velocity over this six-second interval?

   • A) 0 m/s²
   • B) 6 m/s²
   • C) 24 m/s²
   • D) 12 m/s²
6. 6.

   An automated inspection cart travels along a straight track with a velocity described by $v(t) = 3t^2 - 12t + 9\ \text{m/s}$ for $t \ge 0$, where $t$ is in seconds. The cart starts from the origin at $t = 0\ \text{s}$. What is the total distance traveled by the cart between $t = 0\ \text{s}$ and $t = 4\ \text{s}$?

   • A) $12\ \text{m}$
   • B) $4\ \text{m}$
   • C) $8\ \text{m}$
   • D) $16\ \text{m}$
7. 7.

   An autonomous test vehicle travels along a straight calibration track with an acceleration modeled by $a(t) = 3t^2 - 12t\ \text{m/s}^2$ for $t \ge 0$, where $t$ is in seconds. At $t = 0\ \text{s}$, the vehicle is observed at the origin $x(0) = 0\ \text{m}$ moving with an initial velocity of $v(0) = 9\ \text{m/s}$. What is the position of the vehicle along the track at the instant its acceleration returns to zero?

   • A) $-64\ \text{m}$
   • B) $-28\ \text{m}$
   • C) $-23\ \text{m}$
   • D) $-32\ \text{m}$

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