The speed of a train is reduced from 80knhr-1 to 40kmhr-1 in a distance of 500m on applying the brake.how much further will the train travel before coming,how long will it take to bring the train to rest after the application of the brakes?
Well, the train is really pulling the brakes here! Let's solve this question and have a little fun along the way.
To find out how much further the train will travel before coming to a stop, we need to calculate the braking distance. This can be done using the equation:
Braking Distance = Initial Speed^2 / (2 * Braking Deceleration)
Now, we already know the initial speed is 80 km/hr and the final speed (when the train comes to rest) is 0 km/hr. So, let's find out the braking distance!
Braking Distance = (80 km/hr)^2 / (2 * Braking Deceleration)
But hold on, we don't know the braking deceleration yet! So, let's first find the deceleration using the formula:
Deceleration = (Final Speed - Initial Speed) / Time Taken
We know the initial speed is 80 km/hr, the final speed is 0 km/hr, and the distance covered is 500 m. Let's calculate the time taken to bring this train to rest!
Time Taken = Distance / Average Speed
Time Taken = 500 m / ((80 km/hr + 0 km/hr) / 2)
Now that we know the time taken, let's find the deceleration!
Deceleration = (0 km/hr - 80 km/hr) / Time Taken
Once we have the deceleration, we can substitute it back in the braking distance equation to find out how much further the train will travel before coming to rest.
Alright, let's get our math hats on and compute these values!
To find out how much further the train will travel before coming to a stop, we need to calculate the distance covered during the deceleration phase.
Given:
Initial speed, u = 80 km/hr
Final speed, v = 40 km/hr
Distance covered, s = 500 m
First, let's convert the speeds from km/hr to m/s:
Initial speed, u = 80 km/hr = (80 * 1000) m/ (60 * 60) s = 22.22 m/s
Final speed, v = 40 km/hr = (40 * 1000) m/ (60 * 60) s = 11.11 m/s
We can use the equation:
v^2 = u^2 + 2as
Rearranging the equation to calculate the distance, s:
s = (v^2 - u^2) / 2a
Where:
a = acceleration (negative value for deceleration)
u = initial velocity
v = final velocity
Acceleration (deceleration), a = (v - u) / t
We don't know the time, t it takes to come to a stop, but we know the distance s. So we can rewrite the equation for acceleration as:
t = (v - u) / a
Let's calculate the distance covered during the deceleration phase:
s = (v^2 - u^2) / 2a
Plugging in the values:
s = (11.11^2 - 22.22^2) / (2 * a)
Since a = (v - u) / t, we can substitute it back into the distance equation:
s = (11.11^2 - 22.22^2) / (2 * (v - u) / t)
Now, we need to calculate the time it takes to bring the train to rest:
t = (v - u) / a
Plugging in the values:
t = (40 - 80) / (a)
Now, let's solve for a:
a = (v - u) / t = (40 - 80) / (t)
We have two equations:
s = (11.11^2 - 22.22^2) / (2 * (v - u) / t)
and
a = (v - u) / t = (40 - 80) / (t)
Now, we can plug in the known values and solve for s and t.
To find out how much further the train will travel before coming to a stop, we need to calculate the total distance traveled by the train while decelerating from 80 km/hr to 40 km/hr.
First, let's convert the initial speed from km/hr to m/s:
Initial speed = 80 km/hr = (80 * 1000) m / (60 * 60) s ≈ 22.22 m/s
Next, we need to find the acceleration of the train. We can use the equation:
Final speed = Initial speed + (Acceleration * Time)
Since the final speed is 40 km/hr and the initial speed is 80 km/hr, we have:
40 km/hr = 22.22 m/s + (Acceleration * Time)
To simplify further calculations, let's convert the final speed from km/hr to m/s:
Final speed = 40 km/hr = (40 * 1000) m / (60 * 60) s ≈ 11.11 m/s
Now, we can rewrite the equation:
11.11 m/s = 22.22 m/s + (Acceleration * Time)
The distance covered while decelerating can be calculated using the formula:
Distance = Initial velocity * Time + (1/2) * Acceleration * Time^2
Since we want to find the total distance, we can substitute the initial velocity as 22.22 m/s:
Distance = 22.22 m/s * Time + (1/2) * Acceleration * Time^2 ...(1)
To find the acceleration, we can rearrange the equation:
Acceleration = (Final speed - Initial speed) / Time
Acceleration = (11.11 m/s - 22.22 m/s) / Time
Acceleration = -11.11 m/s / Time ...(2)
Combining equations (1) and (2), we have:
Distance = 22.22 m/s * Time + (1/2) * (-11.11 m/s / Time) * Time^2
Distance = 22.22 * Time - (5.55 * Time^2) ...(3)
We know that the distance covered while decelerating is 500 m. Therefore, we can substitute this value into equation (3) and solve for Time.
500 = 22.22 * Time - 5.55 * Time^2
This is a quadratic equation, which can be solved using various methods such as the quadratic formula or factoring.
Once we have the value of Time, we can substitute it back into equation (3) to find the distance traveled by the train before coming to a stop.