Question

When is the particle speeding up? (Enter your answer using interval notation.)

A) (0, 5)
B) [5, [infinity])
C) (-[infinity], 5)
D) (5, [infinity])

Answer 1

The particle is speeding up when its velocity and acceleration are in the same direction. Given that both are negative and increasingly so after t = 5s, the particle is speeding up in the interval (5, ∞).

Step-by-step explanation:

To determine when the particle is speeding up, we must consider both the velocity and acceleration of the particle. A particle is speeding up when its velocity and acceleration are in the same direction. From the given information, we can deduce the following:

• At t = 1s, both velocity (v(1s) = 15 m/s) and acceleration (a(1s) = 10 m/s²) are positive, indicating the particle is speeding up.
• At t = 2s, velocity increases (v(2s) = 20 m/s) but acceleration is zero (a(2s) = 0 m/s²), suggesting constant speed at that moment.
• At t = 3s, velocity decreases (v(3s) = 15 m/s) and acceleration is negative (a(3s) = -10 m/s²), indicating the particle is slowing down as acceleration opposes velocity.
• At t = 5s, velocity is negative (v(5s) = -25 m/s) and acceleration is also negative (a(5s) = -30 m/s²), meaning the particle is speeding up again but in the opposite direction.

Given that the question is when the particle is speeding up, the correct interval must be when the acceleration is increasingly negative after t = 5s. Therefore, the particle is speeding up in the interval (5, ∞), which corresponds to choice D.