calculate the time taken by a train to travel 1000m if its initial speed is 30m/s and it has an acceralation of 0.1m/s
To calculate the time taken by a train to travel 1000m with an initial speed of 30m/s and an acceleration of 0.1m/s², we can use the equation of motion:
s = ut + (1/2)at²
where:
- s is the distance traveled,
- u is the initial velocity,
- a is the acceleration, and
- t is the time taken.
First, let's rearrange the equation to isolate time (t):
s = ut + (1/2)at²
1000 = (30)t + (1/2)(0.1)t²
1000 = 30t + 0.05t²
Next, let's convert the equation to a quadratic equation by moving all terms to one side:
0.05t² + 30t - 1000 = 0
Now we can solve this quadratic equation to find the value of t. We can use the quadratic formula:
t = (-b ± √(b² - 4ac))/(2a)
For our equation, a = 0.05, b = 30, and c = -1000. Plugging in these values, we get:
t = (-30 ± √(30² - 4(0.05)(-1000)))/(2(0.05))
Simplifying further:
t = (-30 ± √(900 + 200))/(0.1)
t = (-30 ± √1100)/0.1
Calculating the square root:
t = (-30 ± 33.166)/0.1
Now, consider both cases:
1. t = (-30 + 33.166)/0.1
t = 3.166/0.1
t ≈ 31.66 seconds
2. t = (-30 - 33.166)/0.1
t = -63.166/0.1
t ≈ -631.66 seconds
Since time cannot be negative in this context, we discard the negative solution. Therefore, the time taken by the train to travel 1000m is approximately 31.66 seconds.