A train starts from rest and accelerates uniformly at 20 m/s² for 10 minutes. Find the velocity acquired by the train if it maintains a constant velocity for 20 minutes. The brakes... A train starts from rest and accelerates uniformly at 20 m/s² for 10 minutes. Find the velocity acquired by the train if it maintains a constant velocity for 20 minutes. The brakes are then applied, and the train is uniformly retarded. It comes to rest in 5 minutes. Draw a velocity time graph and find: (a) total distance traveled, (b) average velocity of the train. Read more

Steps to Solve

1. Calculate the Final Velocity After Acceleration

The train accelerates at $20 , \text{m/s}^2$ for 10 minutes. First, convert minutes to seconds: $$ t = 10 , \text{min} \times 60 , \text{s/min} = 600 , \text{s} $$

Now, use the formula for final velocity: $$ v = u + at $$ where:

• $u = 0 , \text{m/s}$ (initial velocity),
• $a = 20 , \text{m/s}^2$ (acceleration),
• $t = 600 , \text{s}$ (time).

So, $$ v = 0 + (20 \times 600) = 12000 , \text{m/s} $$

2. Calculate the Distance Traveled During Acceleration

Use the formula for distance: $$ s = ut + \frac{1}{2}at^2 $$ Substituting the values: $$ s = 0 \times 600 + \frac{1}{2} \times 20 \times (600)^2 $$ $$ s = \frac{1}{2} \times 20 \times 360000 = 3600000 , \text{m} $$

3. Calculate the Distance Traveled at Constant Velocity

The train maintains a constant velocity of $12000 , \text{m/s}$ for 20 minutes: $$ t = 20 , \text{min} \times 60 , \text{s/min} = 1200 , \text{s} $$

Distance traveled: $$ s = vt = 12000 \times 1200 = 14400000 , \text{m} $$

4. Calculate the Distance Traveled During Deceleration

The train comes to rest in 5 minutes, which is $5 , \text{min} \times 60 , \text{s/min} = 300 , \text{s}$.

The deceleration can be calculated as: $$ a = \frac{\Delta v}{t} = \frac{0 - 12000}{300} = -40 , \text{m/s}^2 $$

Using the distance formula again: $$ s = ut + \frac{1}{2}at^2 $$ $$ s = 12000 \times 300 + \frac{1}{2} \times (-40) \times (300)^2 $$ $$ s = 3600000 - 1800000 = 1800000 , \text{m} $$

5. Calculate Total Distance Traveled

Add all distances: $$ \text{Total distance} = 3600000 + 14400000 + 1800000 = 21600000 , \text{m} $$

6. Calculate Average Velocity

Total time: $$ T = 10 + 20 + 5 = 35 , \text{min} = 2100 , \text{s} $$

Average velocity: $$ \text{Average velocity} = \frac{\text{Total distance}}{\text{Total time}} = \frac{21600000}{2100} \approx 10286.67 , \text{m/s} $$