Question

velocity-time graph

during what interval(s) is the acceleration negative?

○ 25-40 seconds

10-20 seconds

20-30 seconds

20-25 and 30-40 seconds

Explanation:

Step1: Recall acceleration from v-t graph

Acceleration is the slope of the velocity - time (v - t) graph. Mathematically, \(a=\frac{\Delta v}{\Delta t}\), where \(\Delta v = v_{final}-v_{initial}\) and \(\Delta t=t_{final} - t_{initial}\). If the slope is negative, then \(a<0\) (negative acceleration), which means \(\Delta v<0\) (velocity is decreasing) when \(\Delta t > 0\) (time is increasing).

Step2: Analyze each interval

• Interval 10 - 20 seconds: The velocity is constant (horizontal line on v - t graph). So, \(\Delta v=0\), and \(a = 0\) (zero acceleration).
• Interval 20 - 30 seconds: The velocity is decreasing (the graph is a line with negative slope as velocity goes from a positive value towards zero and then negative). So, \(\Delta v=v_{final}-v_{initial}<0\) (since \(v_{final}0\), so \(a=\frac{\Delta v}{\Delta t}<0\) (negative acceleration).
• Interval 25 - 40 seconds: From 25 - 30 seconds, velocity decreases (negative acceleration), but from 30 - 40 seconds, velocity increases (positive acceleration as \(\Delta v>0\) when going from a negative velocity to a less negative or positive velocity? Wait, no, looking at the graph, from 30 - 40 seconds, the velocity is increasing (going from a low point upwards), so slope is positive there. Wait, the option "20 - 30 seconds" is when velocity is decreasing (slope negative), and let's re - check the options. Wait, the options are: 25 - 40, 10 - 20, 20 - 30, 20 - 25 and 30 - 40. Wait, maybe the graph has a peak at 20 seconds, then from 20 - 30 seconds, velocity decreases (slope negative), then from 30 - 40 seconds, velocity increases (slope positive). Wait, but the correct option should be the interval where slope is negative. Let's re - evaluate:

In a velocity - time graph, negative acceleration occurs when the velocity is decreasing (slope of the v - t graph is negative).

• For 10 - 20 seconds: velocity is constant (slope = 0, acceleration = 0).
• For 20 - 30 seconds: velocity is decreasing (slope is negative, so acceleration is negative).
• For 25 - 40 seconds: part of it (25 - 30) is decreasing, part (30 - 40) is increasing.
• For 20 - 25 and 30 - 40: 20 - 25 (if velocity is constant? No, maybe the graph has a flat part from 10 - 20, then decreases from 20 - 30, then decreases more until 30, then increases from 30 - 40. Wait, maybe the correct interval is 20 - 30 seconds as in that interval velocity is decreasing (slope negative), so acceleration is negative.

Answer:

20 - 30 seconds