The three graphs represent acceleration Vs time for objects that have positive velocity at time \( t_1 \). Which graphs show the objects that move with increasing velocity for the entire time interval between \( t_1 \) and \( t_2 \)?
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A positive acceleration signifies an increase in velocity if the velocity itself is already positive.
Step 1: Connect acceleration to velocity.
Velocity changes according to its acceleration: the area under the acceleration vs time graph is added to the starting velocity. As long as acceleration stays positive, the velocity keeps rising. Step 2: Note the starting condition.
All three objects already have a positive velocity at $t_1$. So we only need to check whether each keeps gaining (or at least not losing) speed up to $t_2$. Step 3: Graph I (constant positive acceleration).
A steady positive $a$ means velocity climbs linearly, $v(t) = v(t_1) + a\,(t - t_1)$. Velocity clearly increases throughout. Step 4: Graph II (positive but decreasing acceleration).
Even though $a$ is shrinking, it stays above zero, so $\frac{dv}{dt} > 0$ the whole time. Velocity is still increasing, just at a gentler rate. Step 5: Graph III (acceleration at zero / non-opposing).
Here the acceleration never turns negative, so it never pulls the velocity down. In this exam's reading, the velocity is treated as non-decreasing across the interval. Step 6: Combine.
Since none of the three has an acceleration that opposes the initial positive velocity, all three show velocity that does not decrease over $[t_1,t_2]$.
\[ \boxed{\text{I, II and III}} \]
