Question

A train starts from rest at station A and accelerates at 0.4 m/s^2 for 60 s. Afterwards it travels with a constant velocity for 25 min. It then decelerates at 0.8 m/s^2 until it is brought to rest at station B.

Determine the distance between the stations. ?s^Delta s = _______.

Answer 1

The distance between the two stations is 37.08 km

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Step-by-step explanation:

Given:

`a_1 \:=\:0.4\:m/s²`

`t_1 \:=\:60\:s`

`v_(i1) \:=\:0\:m/s`

`a_2 \:=\:0\:m/s²`

`t_2 \:=\:25\:min\:=\:1500\:s`

`a_3 \:=\:-0.8\:m/s²`

`v_(f3) \:=\:0\:m/s`

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Required:

Distance from Station A to Station B

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Equation:

`a\:=\:(v_f\:-\:v_i)/(t)`

`v_(ave)\:=\:(v_i\:+\:v_f)/(2)`

`v\:=\:(d)/(t)`

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Solution:

Distance when a = 0.4 m/s²

Solve for
`v_(f1)`

`a\:=\:(v_f\:-\:v_i)/(t)`

`0.4\:m/s²\:=\:(v_f\:-\:0\:m/s)/(60\:s)`

`24\:m/s\:=\:v_f\:-\:0\:m/s`

`v_f\:=\:24\:m/s`

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Solve for
`v_(ave1)`

`v_(ave)\:=\:(v_i\:+\:v_f)/(2)`

`v_(ave)\:=\:(0\:m/s\:+\:24\:m/s)/(2)`

`v_(ave)\:=\:12\:m/s`

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Solve for
`d_1`

`v\:=\:(d)/(t)`

`12\:m/s\:=\:(d)/(60\:s)`

`720\:m\:=\:d`

`d_1\:=\:720\:m`

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Distance when a = 0 m/s²

`v_(f1)\:=\:v_(i2)`

`v_(i2)\:=\:24\:m/s`

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Solve for
`v_(f2)`

`a\:=\:(v_f\:-\:v_i)/(t)`

`0\:m/s²\:=\:(v_f\:-\:24\:m/s)/(1500\:s)`

`0\:=\:v_f\:-\:24\:m/s`

`v_f\:=\:24\:m/s`

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Solve for
`v_(ave2)`

`v_(ave)\:=\:(v_i\:+\:v_f)/(2)`

`v_(ave)\:=\:(24\:m/s\:+\:24\:m/s)/(2)`

`v_(ave)\:=\:24\:m/s`

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Solve for
`d_2`

`v\:=\:(d)/(t)`

`24\:m/s\:=\:(d)/(1500\:s)`

`36,000\:m\:=\:d`

`d_2\:=\:36,000\:m`

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Distance when a = -0.8 m/s²

`v_(f2)\:=\:v_(i3)`

`v_(i3)\:=\:24\:m/s`

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Solve for
`v_(f3)`

`a\:=\:(v_f\:-\:v_i)/(t)`

`-0.8\:m/s²\:=\:(0\:-\:24\:m/s)/(t)`

`(t)(-0.8\:m/s²)\:=\:-24\:m/s`

`t\:=\:(-24\:m/s)/(-0.8\:m/s²)`

`t\:=\:30\:s`

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Solve for
`v_(ave3)`

`v_(ave)\:=\:(v_i\:+\:v_f)/(2)`

`v_(ave)\:=\:(24\:m/s\:+\:0\:m/s)/(2)`

`v_(ave)\:=\:12\:m/s`

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Solve for
`d_3`

`v\:=\:(d)/(t)`

`12\:m/s\:=\:(d)/(30\:s)`

`360\:m\:=\:d`

`d_3\:=\:360\:m`

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Total Distance from Station A to Station B

`d\:= \:d_1\:+\:d_2\:+\:d_3`

`d\:= \:720\:m\:+\:36,000\:m\:+\:360\:m`

`d\:= \:37,080\:m`

`d\:= \:37.08\:km`

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Final Answer:

The distance between the two stations is 37.08 km