Which condition indicates that the particle is speeding up?

Which condition indicates that the particle is speeding up?

• A. v(t) and a(t) have the same sign
• B. v(t) and a(t) have opposite signs
• C. v(t) = 0
• D. |v(t)| increasing

Answer: (A)

Speeding up means the speed of the particle is increasing, which happens when velocity and acceleration point in the same direction. In one dimension, that means v and a have the same sign: if you’re moving to the right (positive v) and speeding up, a is also positive; if you’re moving to the left (negative v) and speeding up, a is negative. When they have opposite signs, the acceleration acts to slow you down, so the speed decreases. The moment when v is zero is a special edge case where you can’t determine speeding up just from the signs at that exact instant, but outside that instant the same-sign condition cleanly signals speeding up. That’s why this condition is the best indicator.

Speeding up means the speed of the particle is increasing, which happens when velocity and acceleration point in the same direction. In one dimension, that means v and a have the same sign: if you’re moving to the right (positive v) and speeding up, a is also positive; if you’re moving to the left (negative v) and speeding up, a is negative. When they have opposite signs, the acceleration acts to slow you down, so the speed decreases. The moment when v is zero is a special edge case where you can’t determine speeding up just from the signs at that exact instant, but outside that instant the same-sign condition cleanly signals speeding up. That’s why this condition is the best indicator.