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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.
Given a position function, how do you find when the particle is at rest?
1. Find the velocity function (v(t)) by taking the derivative of the position function (x(t)). 2. Set (v(t) = 0) and solve for (t).
Given a position function, how do you find the acceleration at a specific time?
1. Find the velocity function (v(t)) by taking the derivative of the position function (x(t)). 2. Find the acceleration function (a(t)) by taking the derivative of the velocity function (v(t)). 3. Evaluate (a(t)) at the given time.
How to determine when a particle changes direction?
1. Find (v(t)). 2. Find critical points of (v(t)) by setting (v(t) = 0). 3. Check if (v(t)) changes sign at those critical points.
Given (x(t) = t^3 - 6t^2 + 9t), find when the particle is at rest.
1. (v(t) = 3t^2 - 12t + 9). 2. Set (3t^2 - 12t + 9 = 0). 3. Solve for (t): (t = 1, 3).
Given (x(t) = t^3 - 6t^2 + 9t), find the acceleration at (t = 2).
1. (v(t) = 3t^2 - 12t + 9). 2. (a(t) = 6t - 12). 3. (a(2) = 6(2) - 12 = 0).
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).