Here is the graph of the velocity function of a particle that is in motion.

On what time interval(s) is the particle speeding up? (Recall that speed is the absolute value of velocity and speeding up means increased speed.)

Question

Here is the graph of the velocity function of a particle that is in motion.

On what time interval(s) is the particle speeding up? (Recall that speed is the absolute value of velocity and speeding up means increased speed.)

BlackTom AI · Accepted Answer

To determine when the particle is speeding up, we need to compare the sign of velocity v(t) with the sign of its acceleration a(t) = dv/dt, since speed |v| increases when sign(v) and sign(a) have the same sign. First, consider the interval near t = 0 to the first crossing: the velocity starts at 0 and becomes negative as t increases from 0 toward about t ≈ 1. In this region, v(t) < 0 and the slope of the graph (the acceleration) is also negative (the curve is going downward as it moves to the negative velocity). When both v and a are negative, sign(v) * a > 0, so the speed |v| is increasing. Therefore, the particle is speeding up on this initial segment, which corresponds roughly to (0, 1). Next, examine the interval after the velocity reaches its negative minimum and begins rising toward positive values. The velocity crosses to positive somewhere after t ≈ 1 or 2 and continues increasing up to t ≈ 6, where the velocity reaches a maximum. In this region, v(t) > 0 and a(t) > 0 (the graph is rising), so sign(v) * a > 0, meaning the speed is increasing. Hence, the particle is speeding up on this interval as it goes from about t ≈ 2 up to t ≈ 6. After t ≈ 6, the velocity remains positive but the graph starts to slope downward (a(t) < 0). Here v > 0 and a < 0, so sign(v) * a < 0, and the speed |v| is decreasing. Therefore, the particle is not speeding up on this later interval. Summarizing, the speeding-up intervals are where v and a share the same sign: initially when v < 0 and a < 0, and then when v > 0 and a > 0. Based on the graph, these occur roughly on (0, 1) and (2, 6). Note: The options in the provided data are missing, but the identified speeding-up intervals in standard interpretation match the form (0,1) and (2,6).

Here is the graph of the velocity function of a particle that is in motion. On what time interval(s) is the particle speeding up? (Recall that speed is the absolute value of velocity and speeding up means increased speed.)

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