Calculus: Rates of Change

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?

meters per second Signup and view all the answers

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

Velocity Signup and view all the answers

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

By differentiating the displacement function Signup and view all the answers

What is the relationship between velocity and acceleration?

Acceleration is the derivative of velocity Signup and view all the answers

What is the purpose of the derivative dv/ds?

To find acceleration given velocity and displacement Signup and view all the answers

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

48 meters 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?

-1200/x^3 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^→?

t^2 i^→ + t^3/3 j^→ 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^→?

576i^→ + 1728j^→ meters 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?

9 meters 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?

0.408 meters 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?

0.03J 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?

0.09J Signup and view all the answers

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

9 meters Signup and view all the answers

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

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

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

W = ∫F dx Signup and view all the answers

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

v = u + at Signup and view all the answers

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

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

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

dv/dt = a Signup and view all the answers

What is the concept of proportionality?

A relationship between two quantities where one quantity changes, the other quantity changes in a consistent way 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?

a = -kv^2 Signup and view all the answers

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

Draw a force diagram, set up F = ma, choose the correct expression for a, and solve the differential equation 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?

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

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

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

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

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

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

Power = Force × Velocity Signup and view all the answers

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

0 m/s 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?

dA/dt = -kA 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?

C(t) = C0 e^(-kt) 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?

dT/dt = -k(T - T_env) 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?

T(t) = T0 e^(-kt) + T_env 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?

dC/dt = -k(C - C_desired) 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?

C(t) = C0 e^(-kt) + C_desired 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.