Particle Motion Review (Calculus)

Front

A particle's speed is increasing if:

Back

Acceleration and velocity have the same sign.

Front

A particle's speed is decreasing if:

Back

Velocity and acceleration have different signs.

Front

What is the formula for velocity in terms of position?

Back

Velocity is the derivative of position with respect to time, \( v(t) = \frac{dx}{dt} \).

Front

What does it mean for a particle to be at rest?

Back

A particle is at rest when its velocity is zero.

Front

How do you determine when a particle's acceleration is negative?

Back

Acceleration is negative when the velocity is decreasing.

Front

What is the relationship between acceleration and velocity?

Back

Acceleration indicates the rate of change of velocity.

Front

If a particle's position is given by \( x(t) = 2t^3 - 27t^2 + 84t - 13 \), how do you find when it is at rest?

Back

Set the velocity \( v(t) = \frac{dx}{dt} \) to zero and solve for \( t \).

Front

What is the significance of the second derivative in particle motion?

Back

The second derivative indicates acceleration, which shows how velocity changes over time.

Front

If a particle's acceleration is positive, what can be inferred about its velocity?

Back

If acceleration is positive and velocity is also positive, the speed is increasing.

Front

What is the formula for acceleration in terms of velocity?

Back

Acceleration is the derivative of velocity with respect to time, \( a(t) = \frac{dv}{dt} \).