All Flashcards
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)|)
Given a position function, how do you find when the particle is at rest?
- 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?
- 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?
- 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.
- (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).
- (v(t) = 3t^2 - 12t + 9). 2. (a(t) = 6t - 12). 3. (a(2) = 6(2) - 12 = 0).
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.