Acceleration of a train

The train passes 700 m, braking with an acceleration of -0.15 m/s2. How long does it break, and what is the final speed of the train if the initial was 55 km/h?

Final Answer:

t =  69.5956 s
v2 =  4.8384 m/s

Step-by-step explanation:

s=700 m a=0.15 m/s2 v1​=55 km/h→ m/s=55:3.6  m/s=15.27778 m/s  s = v1​⋅ t − 21​⋅ a⋅ t2 700=15.277777777⋅ t−0.5⋅ 0.15⋅ t2  700=15.277777777⋅ t−0.5⋅ 0.15⋅ t2 0.075000000000045t2−15.278t+700=0  a=0.075;b=−15.278;c=700 D=b2−4ac=15.2782−4⋅0.075⋅700=23.4104938033 D>0  t1,2​=2a−b±D​​=0.1515.28±23.41​​ t1,2​=101.851852±32.256261 t1​=134.108113119 t2​=69.595590574  t>0 v2​>0 t=t2​=69.5956=69.5956 s

Our quadratic equation calculator calculates it.

v2​=v1​−a⋅ t=15.2778−0.15⋅ 69.5956=4.8384 m/s

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Showing 1 comment:
Dr. Math
To solve this problem, we will use the kinematic equations for uniformly accelerated motion. The train is braking, so the acceleration is negative ( a = -0.15 m/s2 ). We are tasked with finding:

1. The time it takes for the train to brake ( t ).
2. The final speed of the train ( v ).

Given Data:


1. Initial speed of the train, u = 55 km/h .
2. Braking acceleration, a = -0.15 m/s2 .
3. Distance traveled while braking, s = 700 m .

Step 1:

Convert Initial Speed to m/s

The initial speed is given in km/h, but we need it in m/s for consistency with the acceleration units. Convert 55 km/h to m/s:

u = 55 × 10003600 = 15.28 m/s.

Step 2:

Use the Kinematic Equation to Find Final Speed

The kinematic equation that relates initial velocity ( u ), final velocity ( v ), acceleration ( a ), and distance ( s ) is:

v2 = u2 + 2as.


Substitute the known values ( u = 15.28 m/s , a = -0.15 m/s2 , s = 700 m ):

v2 = (15.28)2 + 2(-0.15)(700).


Calculate u2 :

(15.28)2 = 233.48.


Calculate 2as :

2(-0.15)(700) = -210.


Now, substitute these values into the equation:

v2 = 233.48 - 210 = 23.48.


Take the square root to find v :

v = √23.48 = 4.85 m/s.

Step 3:

Convert Final Speed to km/h

The final speed is in m/s, but it is often more intuitive to express it in km/h. Convert 4.85 m/s to km/h:

v = 4.85 × 36001000 = 17.46 km/h.

Step 4:

Use the Kinematic Equation to Find Time

The kinematic equation that relates initial velocity ( u ), final velocity ( v ), acceleration ( a ), and time ( t ) is:

v = u + at.


Solve for t :

t = v - ua.


Substitute the known values ( v = 4.85 m/s , u = 15.28 m/s , a = -0.15 m/s2 ):

t = 4.85 - 15.28-0.15.


Calculate the numerator:

4.85 - 15.28 = -10.43.


Now, divide by a :

t = -10.43-0.15 = 69.53 seconds.

Final Answers:


1. The train brakes for:
 
 

69.53 seconds


 


2. The final speed of the train is:
 
 

17.46 km/h


 





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algebranumbersUnits of physical quantitiesthemes, topicsGrade of the word problem

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