Average acceleration of train

1. The AVERAGE acceleration of train A over t=0-30s
= (final velocity - initial velocity)/Δt
= (24-0)/30
= 0.8ms^-2

2.

a.

Average velocity over t=0-30s
= total displacement/time
= area under v-t graph/30
Obviously, the larger the area under the v-t graph, the higher the average velocity.
From the graph, train A has a larger area under the v-t graph than train B (over t=0-30s), and train B has a larger area under the v-t graph than train C.

∴ average velocity over t = 0-30s in ascending order: C < B < A

c.

From the graph, train A has the smallest slope of the v-t graph at t=30s, train B has a higher slope at t=30s, and train C has the highest.
Since the slope of v-t graph represents the acceleration, and train C has the highest slope at t=30s among the three trains, it tells train C has the highest acceleration at t=30s.

∴ acceleration at t=30s: A < B < C

b.

From the graph & result of c., consider at t=30s, the graph of train C escalates faster than that of train B, and the graph of train B grows faster than that of train A. Therefore just after t=30s, train C should gain more velocity than the other two trains.

Velocity just after t=30s: A < B < C

2015-04-27 23:19:50 補充：
Talking about 2.b/c., we can show the answer mathematically.
Consider the velocities of the trains at t=(30+ε)s,
let the velocities of train A, B and C at t=(30+ε)s be v_A, v_B and v_C respectively.

2015-04-27 23:19:55 補充：
From the graph, by estimating & comparing the slope of tangent lines at t=30s, we can get the acceleration (instantaneous rate of change) of velocity of train C > that of train B > that of train A at t=30s, as acceleration is the rate of change of velocity.

2015-04-27 23:20:01 補充：
Consider t=30s to (30+ε)s,
rate of change of velocity (of train A) = (final velocity - initial velocity)/Δt
= (v_A-24)/ε
As ε remains unchanged, the higher the rate of change of velocity, the higher the value of v_A/B/C-24 ⇒ higher value of v at t=(30+ε)s

As a result, the higher the rate of change

2015-04-27 23:20:10 補充：
*As a result, the higher the rate of change of velocity at t=30s, the higher value of velocity at t=(30+ε)s.