Calculus in physics (derivatives and integrals) Transitional Reasoning Download PDF

Questions

1. 1.

   An experimental linear carrier of mass 4 kg starts from rest at $t = 0\text{ s}$ and moves along a frictionless horizontal track. An automated drive system applies a time-dependent forward force given by $F(t) = 48t - 16t^2$, where $F$ is in newtons and $t$ is in seconds, until the applied force drops back to zero. What is the maximum speed attained by the carrier during this motion?

   • A) 18 m/s
   • B) 9 m/s
   • C) 27 m/s
   • D) 72 m/s
2. 2.

   A remote-controlled rover moves along a straight calibration track such that its velocity is given by $v(t) = 3t^2 - 12t + 9$, where $v$ is in meters per second and $t$ is in seconds. The vehicle starts its test run at $t = 0\text{ s}$ and concludes at $t = 4\text{ s}$. Taking into account any reversals in direction, what is the total distance traveled by the rover during this 4-second interval?

   • A) 4 m
   • B) 8 m
   • C) 16 m
   • D) 12 m
3. 3.

   A materials engineer fabricates a non-uniform rod of length $L = 2\text{ m}$ aligned along the x-axis from $x = 0$ to $x = 2\text{ m}$. The linear mass density of the rod is modeled by $\lambda(x) = 3x^2 + 1$, where $\lambda$ is in $\text{kg/m}$ and $x$ is in meters. In physics, the center of mass along the rod is defined as $x_{\text{cm}} = \frac{\int_0^L x\lambda(x)\,dx}{\int_0^L \lambda(x)\,dx}$. What is the position of the center of mass of the rod?

   • A) 1.0 m
   • B) 1.2 m
   • C) 1.4 m
   • D) 1.5 m
4. 4.

   An electrical engineer tests a prototype charging system where the electric current delivered to an energy storage unit is modeled by $I(t) = 12t - 3t^2$, with $I$ in amperes and time $t \ge 0$ in seconds. The charging phase begins at $t = 0$ and terminates automatically when the current drops back to zero. Using the definition of electric current as the rate of charge flow, $I(t) = \frac{dq}{dt}$, what is the total charge transferred to the unit during this charging phase?

   • A) 16 C
   • B) 24 C
   • C) 32 C
   • D) 48 C
5. 5.

   A particle moving along the x-axis is subjected to a conservative force associated with the potential energy function $U(x) = x^3 - 6x^2 + 9x + 4$, where $U$ is in joules and $x$ is in meters. In physics, the conservative force is given by $F(x) = -\frac{dU}{dx}$, and stable equilibrium occurs at a point where the net force is zero and the potential energy has a local minimum. At which position $x$ is the particle in stable equilibrium?

   • A) 1 m
   • B) 3 m
   • C) 2 m
   • D) 4 m
6. 6.

   An engineer analyzes the motion of a research probe moving along a straight test track, where its momentum is modeled by the function $p(t) = 2t^3 - 6t^2 + 15t + 4$, with $p$ expressed in $\text{kg}\cdot\text{m/s}$ and $t \ge 0$ in seconds. In physics, the net force acting on the probe is defined as the time rate of change of momentum, $F_{\text{net}}(t) = \frac{dp}{dt}$. What is the minimum magnitude of the net force acting on the probe during its motion?

   • A) 3 N
   • B) 6 N
   • C) 9 N
   • D) 15 N
7. 7.

   An automated transport cart moves along a straight rail under the influence of a variable net force given by $F(x) = 3x^2 - 8x + 12$, where force is in newtons and position $x$ is in meters. The cart passes the position $x = 1$ m with an initial kinetic energy of 8 J. What is the kinetic energy of the cart when it reaches the position $x = 3$ m?

   • A) 18 J
   • B) 26 J
   • C) 35 J
   • D) 10 J
8. 8.

   A sensor-guided test cart moves along a straight linear track starting from rest at the origin at $t = 0\text{ s}$. Its acceleration is measured as a function of time by $a(t) = 6t - 2$, where $a$ is in $\text{m/s}^2$ and $t$ is in seconds. What is the instantaneous velocity of the cart when it reaches the position $x = 18\text{ m}$?

   • A) 6 m/s
   • B) 16 m/s
   • C) 27 m/s
   • D) 21 m/s