Calculus in physics (derivatives and integrals) Knowledge-Based Download PDF

Questions

1. 1.

   A non-uniform rod of length 3 m lies along the x-axis from $x = 0$ to $x = 3\text{ m}$. Its linear mass density is given by $\lambda(x) = 2 + 6x$, where $\lambda$ is in kilograms per meter and $x$ is in meters. What is the total mass of the rod?

   • A) 6 kg
   • B) 20 kg
   • C) 33 kg
   • D) 60 kg
2. 2.

   The total electric charge that has passed through the cross-section of a wire up to time $t$ is given by $q(t) = 2t^3 - 5t^2 + 4t$, where $q$ is in coulombs and $t$ is in seconds. What is the instantaneous electric current flowing through the wire at $t = 3\text{ s}$?

   • A) 21 A
   • B) 24 A
   • C) 26 A
   • D) 28 A
3. 3.

   The total energy delivered to a device as a function of time is given by $E(t) = 2t^3 + 3t^2 - 4t + 5$, where $E$ is in joules and $t$ is in seconds. What is the instantaneous power delivered to the device at $t = 2\text{ s}$?

   • A) 10 W
   • B) 25 W
   • C) 36 W
   • D) 32 W
4. 4.

   A one-dimensional force given by $F(x) = 3x^2 - 2x + 1$ acts on an object, where $F$ is in newtons and $x$ is in meters. What is the work done by this force as the object moves from $x = 0$ m to $x = 2$ m?

   • A) 6 J
   • B) 9 J
   • C) 10 J
   • D) 18 J
5. 5.

   The linear momentum of an object moving along a straight line is given by $p(t) = 4t^3 - 5t^2 + 2$, where $p$ is in kilogram-meters per second and $t$ is in seconds. What is the net force acting on the object at $t = 2$ s?

   • A) 28 N
   • B) 14 N
   • C) 38 N
   • D) 68 N
6. 6.

   A particle moves along a straight path with a velocity given by $v(t) = 4t^3 - 6t$, where velocity is in meters per second and time $t$ is in seconds. What is the displacement of the particle from $t = 1$ s to $t = 3$ s?

   • A) 56 m
   • B) 54 m
   • C) 52 m
   • D) 48 m
7. 7.

   An object moves along a straight line with an acceleration given by $a(t) = 6t - 4$, where acceleration is in meters per second squared and time $t$ is in seconds. If the initial velocity of the object is 2 m/s at $t = 0$, what is its velocity at $t = 3$ seconds?

   • A) 14 m/s
   • B) 15 m/s
   • C) 21 m/s
   • D) 17 m/s
8. 8.

   A particle moves along a straight line such that its position is given by $x(t) = 2t^3 - 6t^2 + 5$, where position is in meters and time is in seconds. What is the acceleration of the particle at the instant its velocity becomes zero for $t > 0$?

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