Particle Motion

Question 1

How do you determine the velocity of a particle at time t?

Answer 1

The velocity v(t) is the derivative of the position function s(t):
v(t)=s′(t)
It represents the rate of change of position with respect to time.

Question 2

How do you find when a particle is at rest?

Answer 2

A particle is at rest when v(t)=0
Solve v(t)=0 for t, ensuring the times are within the given interval.

Question 3

How do you determine the acceleration of a particle at time t?

Answer 3

The acceleration a(t) is the derivative of the velocity function v(t) or the second derivative of the position function s(t):
a(t)=v′(t)=s′′(t)
It represents the rate of change of velocity with respect to time.

Question 4

How can you tell if a particle is speeding up or slowing down?

Answer 4

Compare the signs of v(t) and a(t):
If v(t) and a(t) have the same sign, the particle is speeding up.
If v(t) and a(t) have opposite signs, the particle is slowing down.

Question 5

How do you find the total distance traveled by a particle on [a,b]

Answer 5

The total distance is found by integrating the absolute value of the velocity: ∫|v(t)| dt

Question 6

How do you determine the displacement of a particle on [a,b]

Answer 6

∫a b v ( t ) d t or s ( b ) − s ( a )

Question 7

What does the sign of v(t) indicate about the particle’s motion?

Answer 7

v(t)>0 : The particle is moving to the right (or upward).
v(t)<0: The particle is moving to the left (or downward).
v(t)=0: The particle is at rest.

Question 8

How do you find the time when the particle changes direction?

Answer 8

A particle changes direction when v(t) changes sign.
Solve v(t)=0, then check intervals to see where v(t) switches from positive to negative or vice versa.

Question 9

How do you find the position of a particle at time t?

Answer 9

If given the velocity
v(t) and an initial position
𝑠(t) use the formula:
x(t) = x(0) + ∫[0, t] v(τ) dτ

Question 10

What does ∣v(t)∣ represent in particle motion?

Answer 10

The magnitude of velocity is the speed of the particle. Speed is always non-negative.

Question 11

How do you determine if a particle is accelerating in the positive or negative direction?

Answer 11

Check the sign of a(t):
a(t)>0: The particle is accelerating in the positive direction.
a(t)<0: The particle is accelerating in the negative direction.