Questions and Answers
What does the phrase 'rate of change of ______' typically model in calculus?
a derivative
Where is the change in time typically located in a derivative representing a rate of change?
at the bottom
What is the derivative of displacement with respect to time, and what does it represent?
ds/dt, velocity
What are the two derivatives that represent acceleration?
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How do you find the maximum height reached by a projectile?
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What is the first step in finding the speed of an object given its height function?
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How do you find the velocity of an object given its height function and a specific time?
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What can be obtained by integrating velocity with respect to time?
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Derive the equation v = u + at
from the given information.
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Derive the equation s = ut + 1/2 at^2
from the given information.
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Derive the equation v^2 = u^2 + 2as
from the given information.
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A particle moves with a variable acceleration given by dv/dt = 100 - v. Find the time taken for the speed of the particle to increase from 25 m/s to 75 m/s.
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Define direct proportionality and give an example.
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Solve the differential equation a = -kv^2
for a car experiencing retardation proportional to the square of its speed.
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Find the general solution to the differential equation dC/dt ∝ (C_desired - C)
.
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Draw a force diagram for an object falling under the influence of gravity and air resistance, where the air resistance is proportional to the velocity of the object.
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Derive the differential equation for an object falling under the influence of gravity and air resistance, where the air resistance is proportional to the velocity of the object.
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Solve the differential equation dv/dt + (k/m)v = g
to find v(t) for an object falling under the influence of gravity and air resistance.
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A 1500 kg car starts from rest and accelerates uniformly. If the engine exerts a constant power of 6000 W, find the velocity of the car after 3 minutes.
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The population of a city grows according to the differential equation dP/dt = kP, where P(t) is the population at time t and k is a positive constant. Solve the differential equation to find P(t).
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A certain chemical reaction is modeled by the differential equation dC/dt = -kC, where C(t) is the concentration of the chemical at time t and k is a positive constant. If the initial concentration is C0, find the time it takes for the concentration to reduce to one-quarter of its initial value.
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The temperature of a room T(t) changes over time according to the differential equation dT/dt = -k(T - T_env), where T_env is the constant temperature of the environment and k is a positive constant. Find the general solution to the differential equation.
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A certain radioactive substance decays according to the differential equation dA/dt = -kA, where A(t) is the amount of the substance at time t and k is a positive constant. If the initial amount is A0, find the time it takes for the substance to decay to half of its initial value.
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The rate at which a chemical concentration C(t) changes in a solution is proportional to the difference between the current concentration and a constant desired concentration C_desired. Find the general solution to the differential equation.
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A 2000 kg car starts from rest and accelerates uniformly. If the engine exerts a constant power of 8000 W, find the velocity of the car after 2 minutes.
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The concentration of a drug in the bloodstream decreases over time according to the differential equation dC/dt = -kC, where C(t) is the concentration of the drug at time t and k is a positive constant. If the initial concentration is C0, find the time it takes for the concentration to reduce to three-quarters of its initial value.
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A particle has acceleration a = 2t i^→ + t^2 j^→
. Find its velocity after t
seconds.
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Find the maximum height of a toy rocket launched at a carnival, given the height function h(t) = 12t^2 + 4t^2
.
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A string has natural length 3m
and elastic constant 5 N/m
. What is the length of the string when it exerts a force of magnitude 30N
?
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Find the rate of change of the force of attraction, F = 600/x^2
, with respect to x
.
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A particle has displacement s = t^3/3 i^→ + t^4/12 j^→
. Find its displacement after 12
seconds.
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Find the work done in extending an open-coiled spring by 0.1m
, given the natural length 0.3m
and elastic constant 6N/m
.
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Find the force of attraction between two objects, given the formula F = 600/x^2
.
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Derive the equation of motion for an object, given its acceleration function a(t)
.
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An elastic string of length 1.5m
and elastic constant 120 N/m
is attached to the ceiling. How far will the string stretch when a 5kg
mass is attached to the end of it?
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Find the work done by a variable force, given the force function F(x)
and the distance x
.
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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
andv 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
Understand how to model rates of change using derivatives and apply calculus to kinematics, including velocity and acceleration. Learn how to integrate and differentiate to solve problems.