Questions and Answers
What is the derivative that represents velocity?
ds/dt
What is the equation that represents Hooke's Law?
F = -k(l - l_o)
What is the work done by a variable force?
W = ∫F dx
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?
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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?
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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.
