A tracks extends a distance of 478 km without a curve. Supposed a train with an initial speed of 72 km/h travels along the entire length of straight track with a uniform acceleration. the train reaches the end of the track in 5 h, 39 min. What is the trains final speed?
To find the train's final speed, we can use the formula:
v = u + at
Where:
v is the final speed
u is the initial speed
a is the acceleration
t is the time taken
Given that the initial speed (u) is 72 km/h and the time taken (t) is 5 hours and 39 minutes, we need to convert the time to hours.
5 hours and 39 minutes = 5 + 39/60 = 5.65 hours
The distance traveled (s) can be found using the formula:
s = ut + 0.5at^2
Since the train travels the entire length of the straight track, the distance traveled (s) is equal to 478 km.
Now we have two equations:
478 = 72(5.65) + 0.5a(5.65)^2 (equation 1)
v = 72 + a(5.65) (equation 2)
We can rearrange equation 1 to solve for the acceleration (a):
478 - 72(5.65) = 0.5a(5.65)^2
478 - 406.8 = 0.5a(31.9225)
71.2 = 15.96125a
a ≈ 4.466 m/s^2
Now we can substitute the acceleration (a) into equation 2 to find the final speed (v):
v = 72 + 4.466(5.65)
v ≈ 96.3 km/h
Therefore, the train's final speed is approximately 96.3 km/h.
To solve this problem, we can use the equation of motion:
\(v_f = v_i + at\)
Where:
\(v_f\) = final velocity (which we need to find)
\(v_i\) = initial velocity (72 km/h)
\(a\) = acceleration (unknown)
\(t\) = time (5 hours and 39 minutes)
First, let's convert the time to hours:
5 hours + (39 minutes / 60 minutes) = 5.65 hours
Now, we need to find the acceleration. We can use the formula:
\(s = ut + \frac{1}{2}at^2\)
Where:
\(s\) = distance (478 km)
\(u\) = initial velocity (72 km/h)
\(a\) = acceleration (unknown)
\(t\) = time (5.65 hours)
To find the acceleration, we need to rearrange the equation:
\(a = \frac{2(s - ut)}{t^2}\)
Now we can substitute the values:
\(a = \frac{2(478 - 72 \cdot 5.65)}{(5.65)^2}\)
Simplifying this equation will give us the acceleration value.
Once we find the acceleration, we can substitute it back into the first equation to find the final velocity:
\(v_f = v_i + at\)
Substituting the values:
\(v_f = 72 + (a \cdot 5.65)\)
Finally, we can solve for the final velocity.