12-09-2011, 10:18 AM
Hi
You can prove it using the equations of motion - the key trick being to eliminate the distance taken to accelerate/ decelerate, which you don't know or want to know!
It's easier to do it in words:
Under constant acceleration, the mean velocity during a period of accn from 0 to V or decn from V to 0 will be V/2.
Therefore, it will take exactly twice as long to cover the same distance, during braking or acceleration, compared to going at full speed.
from equations of motion, time to accelerate = v/a, hence extra time taken to accelerate = v/2a.
This effect is repeated for both the deceleration and acceleration phases, hence Textra = 0.5 * (decn time + accn time) + Tdwell
You can also demonstrate this using a velocity-time graph, the area under this graph is the distance travelled. Comparing graphs for constant speed and station stopping, the difference in area is
V * (0.5 * (decn time + accn time) + Tdwell)
and hence 0.5 * (decn time + accn time) + Tdwell is extra time taken to cover the same distance.
hope this helps
Reuben
You can prove it using the equations of motion - the key trick being to eliminate the distance taken to accelerate/ decelerate, which you don't know or want to know!
It's easier to do it in words:
Under constant acceleration, the mean velocity during a period of accn from 0 to V or decn from V to 0 will be V/2.
Therefore, it will take exactly twice as long to cover the same distance, during braking or acceleration, compared to going at full speed.
from equations of motion, time to accelerate = v/a, hence extra time taken to accelerate = v/2a.
This effect is repeated for both the deceleration and acceleration phases, hence Textra = 0.5 * (decn time + accn time) + Tdwell
You can also demonstrate this using a velocity-time graph, the area under this graph is the distance travelled. Comparing graphs for constant speed and station stopping, the difference in area is
V * (0.5 * (decn time + accn time) + Tdwell)
and hence 0.5 * (decn time + accn time) + Tdwell is extra time taken to cover the same distance.
hope this helps
Reuben