An airport tram carries passengers between two terminal buildings 500 m apart. The tram starts from rest and accelerations at 1.5 m/s^2 for 20 m. It then travels at a constant velocity for 450 m. Finally, it slows down for 30 m at an acceleration of 1 m/s^2 and comes to rest. How long does it take the tram to travel the entire 500 m trip between buildings?
To find the time it takes for the tram to travel the entire 500 m trip, we can split the journey into three parts: the period of acceleration, the period of constant velocity, and the period of deceleration.
1. Acceleration phase:
The tram starts from rest and accelerates at 1.5 m/s^2 for 20 m. We can use the equation for acceleration to find the time taken during this phase.
a = (v - u) / t
where:
a = acceleration (1.5 m/s^2)
v = final velocity (unknown)
u = initial velocity (0 m/s)
t = time (unknown)
Since the initial velocity is 0 m/s, the equation simplifies to:
a = v / t
Rearranging the equation, we get:
v = a * t
Substituting the values, we have:
1.5 = v / t
Now we can solve for v by multiplying both sides of the equation by t:
1.5t = v
2. Constant velocity phase:
After the acceleration phase, the tram travels at a constant velocity for 450 m. The time taken during this phase can be found using the equation for velocity:
v = d / t
where:
v = velocity (unknown)
d = distance (450 m)
t = time (unknown)
Rearranging the equation, we have:
t = d / v
Substituting the values, we get:
t = 450 / v
3. Deceleration phase:
In this final phase, the tram slows down at an acceleration of 1 m/s^2 for 30 m. We can use the same equation as in the first phase to find the time taken during this phase.
a = v / t
Substituting the values, we have:
1 = v / t
Solving for t:
t = v
Now, we can add up the times for each phase to find the total time taken for the entire journey:
Total time = (time for acceleration phase) + (time for constant velocity phase) + (time for deceleration phase)
Total time = (1.5t) + (450 / v) + t
Since the initial velocity is 0 m/s, the final velocity is also 0 m/s at the end of the deceleration phase. Using this information, we have:
Total time = (1.5t) + (450 / v) + t
Total time = (1.5t) + (450 / (1.5t)) + t
Total time = (2.5t) + (450 / (1.5t))
Substituting the total distance of 500 m, we have:
Total time = (2.5t) + (450 / (1.5t)) = 500 / v
Now we can solve this equation to find the value of t, and then substitute it back into the equation to find the total time taken.
To find the total time it takes for the tram to travel the entire 500 m trip between the two buildings, we need to break down the trip into its different segments and calculate the time for each segment.
Let's calculate the time for each segment of the trip.
Segment 1: Accelerating from rest for 20 m at an acceleration of 1.5 m/s^2
Using the equation of motion: v^2 = u^2 + 2as,
where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and s is the distance traveled.
Here, u = 0, a = 1.5 m/s^2, and s = 20 m.
Plugging in these values, we get:
v^2 = 0^2 + 2 * 1.5 * 20
v^2 = 0 + 60
v^2 = 60
v = √60
v ≈ 7.745 m/s
To find the time taken for this segment, we use the equation: t = (v - u) / a.
Plugging in the values, we get:
t = (7.745 - 0) / 1.5
t ≈ 5.163 seconds
Segment 2: Traveling at a constant velocity of 450 m.
Since the tram is traveling at a constant velocity, the time taken for this segment is simply the distance divided by the velocity:
t = 450 / v
t = 450 / 7.745
t ≈ 58.126 seconds
Segment 3: Slowing down for 30 m at an acceleration of 1 m/s^2.
Using the same equation of motion:
0 = v^2 + 2 * (-1) * 30
v^2 = -60
v = √(-60) (ignoring the negative square root since we're dealing with magnitudes)
v ≈ 7.746 m/s
Using the equation t = (v - u) / a, where u is the initial velocity (7.745 m/s) and a is the acceleration (-1 m/s^2), we can find the time taken for this segment:
t = (0 - 7.746) / -1
t ≈ 7.746 seconds
Now, we can find the total time by summing up the times for each segment:
Total time = time for segment 1 + time for segment 2 + time for segment 3
Total time ≈ 5.163 + 58.126 + 7.746
Total time ≈ 71.035 seconds
Therefore, it takes approximately 71.035 seconds for the tram to travel the entire 500 m trip between the two buildings.