Calculus: Rates of Change
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Questions and Answers

What does the rate of change of height represent?

  • Displacement
  • Velocity
  • Acceleration
  • Derivative of displacement with respect to time (correct)

    • What is the derivative of velocity with respect to displacement?

  • Deceleration
  • dv/dt
  • v dv/ds (correct)
  • Acceleration

    • How can you find the maximum height reached by an object?

  • When the displacement is maximum
  • When the time is maximum
  • When the acceleration is maximum
  • When the velocity is zero (correct)

    • What is the unit of velocity when it is represented as ds/dt?

    <p>meters per second</p> Signup and view all the answers

    What does the derivative of displacement with respect to time represent?

    <p>Velocity</p> Signup and view all the answers

    How can you find the velocity of an object given its displacement function?

    <p>By differentiating the displacement function</p> Signup and view all the answers

    What is the relationship between velocity and acceleration?

    <p>Acceleration is the derivative of velocity</p> Signup and view all the answers

    What is the purpose of the derivative dv/ds?

    <p>To find acceleration given velocity and displacement</p> Signup and view all the answers

    What is the height of the toy rocket after 3 seconds?

    <p>48 meters</p> Signup and view all the answers

    What is the rate of change of F with respect to x in the formula F = 600/x^2?

    <p>-1200/x^3</p> Signup and view all the answers

    What is the velocity of the particle after t seconds, given the acceleration a = 2t i^→ + t^2 j^→?

    <p>t^2 i^→ + t^3/3 j^→</p> Signup and view all the answers

    What is the displacement of the particle after 12 seconds, given the acceleration a = 2t i^→ + t^2 j^→?

    <p>576i^→ + 1728j^→ meters</p> Signup and view all the answers

    What is the length of the string when it exerts a force of magnitude 30N, given the elastic constant k = 5 N/m and natural length l_o = 3m?

    <p>9 meters</p> Signup and view all the answers

    How far will the string stretch when a 5kg mass is attached to the end of it, given the elastic constant k = 120 N/m and natural length l_o = 1.5m?

    <p>0.408 meters</p> Signup and view all the answers

    What is the work done in extending the spring by 0.1m, given the natural length 0.3m and elastic constant 6N/m?

    <p>0.03J</p> Signup and view all the answers

    What is the work done in extending the spring from 0.1m to 0.2m, given the natural length 0.3m and elastic constant 6N/m?

    <p>0.09J</p> Signup and view all the answers

    What is the maximum height of an object, given the function H(t) = 6t - t^2?

    <p>9 meters</p> Signup and view all the answers

    What is the derivative of acceleration used to derive the SUVAT equations?

    <p>dv/dt, v dv/ds, and ds/dt</p> Signup and view all the answers

    What is the formula for the work done by a variable force?

    <p>W = ∫F dx</p> Signup and view all the answers

    What is the correct equation for the velocity of an object under constant acceleration?

    <p>v = u + at</p> Signup and view all the answers

    What is the correct equation for the displacement of an object under constant acceleration?

    <p>s = ut + 1/2 at^2</p> Signup and view all the answers

    What is the correct equation for the acceleration of an object under variable acceleration?

    <p>dv/dt = a</p> Signup and view all the answers

    What is the concept of proportionality?

    <p>A relationship between two quantities where one quantity changes, the other quantity changes in a consistent way</p> Signup and view all the answers

    What is the correct equation for the retardation of a car experiencing a retardation proportional to the square of its speed?

    <p>a = -kv^2</p> Signup and view all the answers

    What is the correct approach to solve a problem involving variable forces?

    <p>Draw a force diagram, set up F = ma, choose the correct expression for a, and solve the differential equation</p> Signup and view all the answers

    What is the correct equation for the velocity of an object under the influence of gravity and air resistance?

    <p>dv/dt = g - kv^2</p> Signup and view all the answers

    What is the correct approach to solve a problem involving proportional acceleration?

    <p>Write the proportionality as an equation, and then integrate to find the solution</p> Signup and view all the answers

    What is the correct equation for the displacement of an object under variable acceleration?

    <p>s = ut + ∫(1/2) a dt</p> Signup and view all the answers

    What is the formula for power in terms of force and velocity?

    <p>Power = Force × Velocity</p> Signup and view all the answers

    If an object starts from rest, what is the initial velocity?

    <p>0 m/s</p> Signup and view all the answers

    What is the differential equation for the amount of a radioactive substance A(t) in terms of time t?

    <p>dA/dt = -kA</p> Signup and view all the answers

    What is the general solution to the differential equation dC/dt = -kC, where C(t) is the concentration of a drug at time t?

    <p>C(t) = C0 e^(-kt)</p> Signup and view all the answers

    What is the differential equation for the temperature of a cup of coffee T(t) in terms of time t, according to Newton's Law of Cooling?

    <p>dT/dt = -k(T - T_env)</p> Signup and view all the answers

    What is the general solution to the differential equation dT/dt = -k(T - T_env), where T0 is the initial temperature of the coffee?

    <p>T(t) = T0 e^(-kt) + T_env</p> Signup and view all the answers

    What is the rate at which a chemical concentration C(t) changes in a solution, according to the differential equation?

    <p>dC/dt = -k(C - C_desired)</p> Signup and view all the answers

    What is the general solution to the differential equation dC/dt = -k(C - C_desired), where C_desired is the desired concentration?

    <p>C(t) = C0 e^(-kt) + C_desired</p> Signup and view all the answers

    Study Notes

    Rates of Change

    • When a question mentions "rate of change of ______", we can model it as a derivative.
    • The rate of change of time will always be at the bottom of the derivative, unless stated otherwise in the question.

    Calculus in Kinematics

    • Velocity (v) is the derivative of displacement (s) with respect to time (t): ds/dt = v
    • Acceleration (a) is the derivative of velocity with respect to time: dv/dt = a
    • We can integrate velocity to find displacement and differentiate acceleration to find velocity.

    Worked Examples in Kinematics

    • Example 1: Finding the height, speed, and maximum height of a ball thrown vertically upwards using derivatives.
    • Example 2: Finding the rate of change of the force of attraction between two objects using derivatives.

    Vector Calculus

    • To differentiate or integrate a vector, do it separately for the i and j components.
    • Worked Example 1: Finding the velocity and displacement of a particle with acceleration a = 2t i + t^2 j.

    Hookes Law

    • Hookes Law states that the restoring force (F) is proportional to the extension (l-lo) of a spring or string: F = -k(l-lo)
    • The "minus" sign indicates that the restoring force is in the opposite direction.
    • Worked Examples 1 and 2: Finding the length of a string when it exerts a certain force using Hookes Law.

    Work Done by a Variable Force

    • The formula for work done is W = ∫F dx.
    • Worked Example 1: Finding the work done in extending a spring by a certain distance using Hookes Law.

    Deriving Equations of Motion

    • We can derive the equations of motion using calculus, including:
      • v = u + at (velocity-time equation)
      • s = ut + 1/2 at^2 (position-time equation)
      • v^2 = u^2 + 2as (velocity-position equation)
    • Derivations involve setting up equations using the derivatives dv/dt = a and v dv/ds = a.

    Variable Acceleration

    • If acceleration is linked to velocity and time, use the derivative dv/dt = a.
    • If acceleration is linked to velocity and displacement, use the derivative v dv/ds = a.
    • Worked Example 1: Finding the time taken for the speed of a particle to increase from 25 m/s to 75 m/s using a differential equation.

    Proportional Acceleration

    • There are two types of proportionality: direct and inverse.
    • Direct proportionality: y = kx, where k is the constant of proportionality.
    • Inverse proportionality: y = k/x, where k is the constant of proportionality.
    • Worked Example 1: Finding the time taken for a car to travel a certain distance using a proportionality.

    Non-Mechanics Calculus

    • These questions involve solving differential equations, such as:
      • dA/dt = -kA ( radioactive substance decay)
      • dC/dt = -kC (concentration of a drug in the bloodstream)
      • dT/dt = -k(T - T_env) (Newton's Law of Cooling)
    • Worked Examples 1 and 2: Solving these differential equations to find the general solutions.

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    Description

    Learn how to model rates of change as derivatives, understanding the concept of dh/dt and its application in kinematics.

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