a train slows down from 104m/s to 76m/s with constant decceleration in 440m. 1) what is the acceleration of the train? 2) how long does it take the train to travel this distance? 3) plot the position, velocity and acceleration of the train during this time interval as a function of time, using the initial position of the train as the origin.
Vf^2=Vi^2+2ad solve for a
then 2) vf=vi+at solve for t.
To solve these questions, we can use the equations of motion for uniformly decelerating motion. These equations relate the initial and final velocities, acceleration, time, and displacement of an object.
1) What is the acceleration of the train?
We can use the equation:
v^2 = u^2 + 2as
where:
v = final velocity = 76 m/s
u = initial velocity = 104 m/s
a = acceleration (to be determined)
s = displacement = 440 m
Rearranging the equation and plugging in the known values, we get:
a = (v^2 - u^2) / (2s)
a = (76^2 - 104^2) / (2 * 440)
a = 5776 - 10816 / 880
a = -5040 / 880
a = -5.73 m/s^2 (rounded to two decimal places)
Therefore, the acceleration of the train is approximately -5.73 m/s^2.
2) How long does it take the train to travel this distance?
We can use the equation:
v = u + at
where:
v = final velocity = 76 m/s
u = initial velocity = 104 m/s
a = acceleration = -5.73 m/s^2 (from the previous calculation)
t = time (to be determined)
Rearranging the equation and plugging in the known values, we have:
t = (v - u) / a
t = (76 - 104) / -5.73
t = -28 / -5.73
t = 4.88 seconds (rounded to two decimal places)
Therefore, it takes approximately 4.88 seconds for the train to travel the distance of 440 m.
3) Plotting the position, velocity, and acceleration of the train as a function of time:
To plot these quantities, we need the equations that relate them to time.
Position:
We can use the equation:
s = ut + (1/2)at^2 + s0
where:
s = displacement (position)
u = initial velocity
a = acceleration
t = time
s0 = initial position (taken as the origin)
In this case, we assume the train is initially at the origin, so s0 = 0.
The equation simplifies to:
s = ut + (1/2)at^2
Velocity:
We can use the equation:
v = u + at
Acceleration:
We can use the previously calculated value of acceleration: a = -5.73 m/s^2.
With these equations, we can now create the plots of position, velocity, and acceleration as functions of time.