An electric train accelerates uniformly from rest at a rate of 2\text{ m/s}^2 and then imm… — JAMB UTME

An electric train accelerates uniformly from rest at a rate of $2\text{ m/s}^2$ and then immediately decelerates uniformly at $4\text{ m/s}^2$ until it comes to a complete stop. If the total distance covered by the train during this entire motion is $600\text{ m}$ , what is the total time taken for the journey?

A
$20\text{ s}$
B
$25\text{ s}$
$30\text{ s}$ Answer
D
$40\text{ s}$

Answer

The total time taken for the journey is

30\text{ s}

For motion from rest back to rest, the velocity-time graph forms a triangle of total base

T

and height

v_{\text{max}}

. Acceleration time is

t_1 = v_{\text{max}}/2

and deceleration time is

t_2 = v_{\text{max}}/4 = 0.5 t_1

. The total displacement is

s = \frac{1}{2} (2) t_1^2 + \frac{1}{2} (4) t_2^2 = t_1^2 + 2 (0.5 t_1)^2 = 1.5 t_1^2 = 600\text{ m}

, yielding

t_1 = 20\text{ s}

and

t_2 = 10\text{ s}

. Summing these gives a total journey time of

30\text{ s}

Step-by-Step Solution

Relate maximum velocity to time in each stage.

Let

v_{\text{max}}

be the maximum velocity. Acceleration time

t_1 = \frac{v_{\text{max}}}{2}

and deceleration time

t_2 = \frac{v_{\text{max}}}{4}

Using

v = u + at

from rest to

v_{\text{max}}

and from

v_{\text{max}}

to rest.

Express total time

T

in terms of maximum velocity.

T = t_1 + t_2 = \frac{v_{\text{max}}}{2} + \frac{v_{\text{max}}}{4} = \frac{3}{4} v_{\text{max}}

, which gives

v_{\text{max}} = \frac{4}{3} T

Total duration is the sum of individual phase durations.

Set up distance equation from the area of the velocity-time graph.

\text{Total distance } s = \frac{1}{2} \times T \times v_{\text{max}} = \frac{1}{2} \times T \times \frac{4}{3} T = \frac{2}{3} T^2 = 600\text{ m}.

Area under a velocity-time triangle equals total displacement.

Solve for total time

T

T^2 = 600 \times \frac{3}{2} = 900 \implies T = 30\text{ s}

Taking the square root gives the total travel time.

Key Concept

Multi-stage linear motion with uniform acceleration and deceleration

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