Podcast
Questions and Answers
What is the derivative that represents velocity?
What is the derivative that represents velocity?
ds/dt
What is the equation that represents Hooke's Law?
What is the equation that represents Hooke's Law?
F = -k(l - l_o)
What is the work done by a variable force?
What is the work done by a variable force?
W = ∫F dx
What is the derivative that represents acceleration?
What is the derivative that represents acceleration?
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What is the equation that represents the relationship between velocity and acceleration when acceleration is constant?
What is the equation that represents the relationship between velocity and acceleration when acceleration is constant?
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How do you differentiate or integrate a vector?
How do you differentiate or integrate a vector?
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What is the type of proportionality when one quantity changes, and the other quantity changes in a consistent way?
What is the type of proportionality when one quantity changes, and the other quantity changes in a consistent way?
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When converting a proportionality (∝) into an equation (=), what is important to add?
When converting a proportionality (∝) into an equation (=), what is important to add?
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Study Notes
Rate of Change
- When a question asks for the "rate of change of ______", model it as a derivative.
- The rate of change of time is always at the bottom unless stated otherwise in the question.
Velocity and Acceleration
- Velocity as a derivative is ds/dt.
- Acceleration has two derivatives: dv/dt and v dv/ds.
- We can integrate dv/dt to find velocity.
- We use v dv/ds when acceleration is mentioned but is expressed using velocity and displacement.
Vector Calculus
- When differentiating or integrating a vector, treat the i and j parts separately.
Hooke's Law
- Hooke's Law states that the restoring force (F) is proportional to the extension (l - l₀) of a spring/string, and is expressed as F = -k(l - l₀).
Work Done by a Variable Force
- Work is force multiplied by distance.
- When the force is changing, adjust the formula by adding an integral: W = ∫F dx.
- Work is energy, and its units are Joules (J).
Derivations
- Derivation 1: v = u + at, where v is final velocity, u is initial velocity, a is acceleration, and t is time.
- Derivation 2: s = ut + 1/2 at², where s is displacement, u is initial velocity, a is acceleration, and t is time.
- Derivation 3: v² = u² + 2as, where v is final velocity, u is initial velocity, a is acceleration, and s is displacement.
Variable Acceleration
- When linking acceleration to velocity and time, use the derivative dv/dt = a.
- When linking velocity and displacement to acceleration, use v dv/ds = a.
Proportional Acceleration
- Proportionality is a relationship between two quantities where if one quantity changes, the other quantity changes consistently.
- There are two main types of proportionality: direct proportionality (x ∝ y) and inverse proportionality (x ∝ 1/y).
- When converting a proportionality to an equation, always add a constant to ensure validity.
Variable Forces
- To solve variable force questions:
- Draw a force diagram.
- Set up F = ma.
- Choose the correct expression for a.
- Solve the differential equation.
Power
- The formula for power is Power = Force x Velocity.
Non-Mechanics Calculus
- For non-mechanics calculus questions, the differential equations are given, and you simply need to solve them.
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Description
This quiz covers key concepts in calculus, including rates of change, velocity, acceleration, and vector calculus. It provides formulas and rules for modeling and solving problems.
