Calculus · Calculus — University

Video solutions

• Graphing Position, Velocity, and Acceleration for Sinusoidal Motion
  Calculus · Calculus — University

  A particle's position is defined by $s = [2\sin(\pi t/5) + 4]$ m, where $t$ is in seconds. Construct the $s$–$t$, $v$–$t$, and $a$–$t$ graphs for $0 \leq t \leq 10$ s.

• Constructing v–t and a–t Graphs from a Velocity Function
  Calculus · Calculus — University

  A particle starts from $s = 0$ and travels along a straight line with a velocity $v = (t^2 - 4t + 3)$ m/s, where $t$ is in seconds. Construct the $v$–$t$ and $a$–$t$ graphs for the time interval $0 \le t \le 4$ s.

• Escape Velocity via Separation of Variables
  Calculus · Calculus — University

  As a body is projected to a high altitude above the Earth's surface, the variation of the acceleration of gravity with respect to altitude $y$ must be taken into account. Neglecting air resistance, the acceleration is gi…

• Kinematics with Variable Acceleration: Finding Distance Traveled
  Calculus · Calculus — University

  A sphere is fired downwards into a medium with an initial speed of 27 m/s. It experiences a deceleration of $a = -6t$ m/s², where $t$ is in seconds. Determine the distance traveled before the sphere stops.

• Integration of Motion Equations: Position, Distance, and Velocity
  Calculus · Calculus — University

  The acceleration of a particle along a straight line is defined by $a = (2t - 9)$ m/s², where $t$ is in seconds. At $t = 0$, $s = 1$ m and $v = 10$ m/s. When $t = 9$ s, determine (a) the particle's position, (b) the tota…

• Particle Velocity from a Position-Dependent Acceleration
  Calculus · Calculus — University

  The acceleration of a particle traveling along a straight line is $a = \dfrac{1}{4}\,s^{1/2}$ m/s², where $s$ is in meters. If $v = 0$ and $s = 1$ m when $t = 0$, determine the particle's velocity at $s = 2$ m.

• Kinematics with Exponential Velocity: Distance and Acceleration
  Calculus · Calculus — University

  A freight train travels at $v = 60\left(1 - e^{-t}\right)$ ft/s, where $t$ is the elapsed time in seconds. Determine the distance traveled in three seconds, and the acceleration at this time.

• Kinematics with Calculus: Position, Distance, and Acceleration
  Calculus · Calculus — University

  The velocity of a particle traveling along a straight line is $v = 3t^2 - 6t$ ft/s, where $t$ is in seconds. If $s = 4$ ft when $t = 0$, determine the position of the particle when $t = 4$ s. What is the total distance t…

• Rocket Altitude via Kinematics and Integration
  Calculus · Calculus — University

  The acceleration of a rocket traveling upward is given by $a = (6 + 0.02s)$ m/s², where $s$ is in meters. Determine the time needed for the rocket to reach an altitude of $s = 100$ m. Initially, $v = 0$ and $s = 0$ when …

• Rectilinear Kinematics: Distance, Velocity, and Acceleration
  Calculus · Calculus — University

  The position of a particle along a straight-line path is defined by $$s(t) = t^3 - 6t^2 - 15t + 7 \text{ ft}$$ where $t$ is in seconds. Determine the total distance traveled when $t = 10$ s. Find the particle's average v…

• Kinematics with Polynomial Position Functions
  Calculus · Calculus — University

  A particle moves along a straight line with position $s = (10t^2 + 20)$ mm. Find: (a) Displacement from $t = 1$ s to $t = 5$ s (b) Average velocity over that interval (c) Acceleration at $t = 1$ s

• Position, Displacement, and Total Distance Using Derivatives
  Calculus · Calculus — University

  Given the position function $s = 1.5t^3 - 13.5t^2 + 22.5t$ ft, where $t$ is in seconds, find: 1. The position at $t = 6$ s 2. The total distance traveled from $t = 0$ to $t = 6$ s

• Velocity as a Function of Position Under Constant Acceleration
  Calculus · Calculus — University

  A particle travels along a straight line with constant acceleration. Given: when $s = 4$ ft, $v = 3$ ft/s; when $s = 10$ ft, $v = 8$ ft/s. Find $v$ as a function of $s$.

• Kinematics via Differentiation and Integration
  Calculus · Calculus — University

  Given: $v = (6t - 3t^2)$ m/s, $s = 0$ when $t = 0$. Find at $t = 3$ s: 1. **Deceleration (acceleration)** 2. **Position $s$** 3. **Total distance traveled** 4. **Average speed**

• Kinematics with Calculus: Velocity, Acceleration, and Distance
  Calculus · Calculus — University

  A particle moves along a straight line with velocity $v = (12 - 3t^2)$ m/s. At $t = 1$ s, the position is $x = -10$ m (10 m to the left of the origin). Find: 1. The acceleration at $t = 4$ s 2. The displacement from $t =…

• Kinematics: Position Under Constant Acceleration
  Calculus · Calculus — University

  A particle has an initial velocity of $v_0 = +12 \text{ ft/s}$ (to the right), an initial position of $s_0 = 0$, and a constant acceleration of $a = -2 \text{ ft/s}^2$ (to the left). Find the position $s$ of the particle…

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