How do you determine the time at which a particle is at rest?

How do you determine the time at which a particle is at rest?

• A. a(t) = 0
• B. |v(t)| = 0
• C. v(t) = 0
• D. ∫ v(t) dt = 0

Answer: (C)

Rest is defined by the instantaneous velocity being zero—the particle isn’t changing its position at that exact moment. So, to find when it’s at rest, you look for times t where the velocity function v(t) equals zero and solve v(t) = 0 for t. Why not the other ideas? If acceleration a(t) is zero, the velocity isn’t necessarily zero at that moment; it just means velocity isn’t changing at that instant. A particle could be at rest at one moment and then start moving if the acceleration is nonzero there. The integral of velocity over time gives displacement, not the instantaneous state; having zero net displacement over an interval doesn’t guarantee the particle was at rest at any particular time within that interval. The speed |v(t)| being zero would also indicate rest (speed is the magnitude of velocity), but the direct condition you solve for is v(t) = 0.

Rest is defined by the instantaneous velocity being zero—the particle isn’t changing its position at that exact moment. So, to find when it’s at rest, you look for times t where the velocity function v(t) equals zero and solve v(t) = 0 for t.

Why not the other ideas? If acceleration a(t) is zero, the velocity isn’t necessarily zero at that moment; it just means velocity isn’t changing at that instant. A particle could be at rest at one moment and then start moving if the acceleration is nonzero there. The integral of velocity over time gives displacement, not the instantaneous state; having zero net displacement over an interval doesn’t guarantee the particle was at rest at any particular time within that interval. The speed |v(t)| being zero would also indicate rest (speed is the magnitude of velocity), but the direct condition you solve for is v(t) = 0.