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What is the formula for velocity given position (x(t))?
\(v(t) = \frac{dx}{dt})
What is the formula for acceleration given velocity (v(t))?
\(a(t) = \frac{dv}{dt})
What is the formula for acceleration given position (x(t))?
\(a(t) = \frac{d^2x}{dt^2})
How is speed calculated?
Speed = (|v(t)|)
Define velocity in the context of motion.
Velocity is the rate of change of position with respect to time: (v(t) = \frac{dx}{dt}).
Define acceleration in the context of motion.
Acceleration is the rate of change of velocity with respect to time: (a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}).
What is 'speed' in relation to velocity?
Speed is the magnitude (absolute value) of velocity: (|v(t)|).
Define instantaneous rate of change.
The instantaneous rate of change is the derivative of a function at a specific point.
What does positive velocity indicate?
Movement in the positive direction (e.g., to the right).
What does negative velocity indicate?
Movement in the negative direction (e.g., to the left).
What does zero acceleration mean?
Constant velocity.
What does positive acceleration mean?
Speeding up (velocity and acceleration have the same sign).
What does negative acceleration mean?
Slowing down (velocity and acceleration have opposite signs).
Define position function.
A function, often denoted (x(t)), that gives the location of an object at time (t).
Explain the relationship between position, velocity, and acceleration.
Velocity is the derivative of position, and acceleration is the derivative of velocity. (x(t) \rightarrow v(t) \rightarrow a(t))
What does the sign of the velocity tell you about the motion of a particle?
The sign of the velocity indicates the direction of motion. Positive means moving in the positive direction, negative means moving in the negative direction.
How do you determine if a particle is speeding up or slowing down?
Compare the signs of velocity and acceleration. If they are the same, the particle is speeding up; if they are different, it is slowing down.
Explain the meaning of the derivative in the context of motion.
The derivative represents the instantaneous rate of change. In motion, it connects position, velocity, and acceleration.