A bus accelerates at 1.4 m/s2 from rest for 14 s. It then travels at constant speed for 23 s, after which it slows to a stop with an acceleration of magnitude 1.3 m/s2.
a. V = a*t = 1.4 * 14 = 19.6 m/s.
d = 0.5a*t^2 + V*t2 + (Vf-V)/2a
b. d = 0.5*1.4*14^2 + 19.6*23 + (0-(19.6^2)/-2.6 = 137.2 + 450.8 + 147.8
= 736 m.
Sure! I can help you with that. Let's break down the motion of the bus into three stages:
Stage 1: Acceleration from rest
Given:
Initial velocity (u) = 0 m/s (rest)
Acceleration (a) = 1.4 m/s^2
Time (t) = 14 s
To find the final velocity (v) after the acceleration, we can use the equation:
v = u + a*t
Substituting the given values:
v = 0 + 1.4 * 14
Calculating:
v = 19.6 m/s
So, after 14 seconds, the bus has a velocity of 19.6 m/s.
Stage 2: Constant speed
The bus continues to travel at a constant speed for 23 seconds, which means there is no acceleration. The velocity remains at 19.6 m/s during this time.
Stage 3: Deceleration to rest
Given:
Initial velocity (u) = 19.6 m/s
Acceleration (a) = -1.3 m/s^2 (negative sign denotes deceleration)
Final velocity (v) = 0 m/s (rest)
We can use the same equation as before to find the time it takes for the bus to come to stop:
v = u + a*t
Substituting the given values:
0 = 19.6 + (-1.3) * t
Rearranging the equation:
1.3t = 19.6
Solving for t:
t = 19.6 / 1.3
Calculating:
t ≈ 15.08 s
So, it takes approximately 15.08 seconds for the bus to come to a stop.
To summarize:
1. In the first 14 seconds, the bus accelerates from rest to a velocity of 19.6 m/s.
2. It then travels at a constant speed of 19.6 m/s for 23 seconds.
3. Finally, it decelerates and comes to a stop in approximately 15.08 seconds.
To find the distance traveled by the bus, we need to break down the motion into three parts: initial acceleration, constant velocity, and final deceleration.
1. Initial acceleration:
The bus starts from rest and accelerates at 1.4 m/s^2 for 14 seconds. To find the distance traveled during this phase, we can use the following equation of motion:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the initial velocity is 0 m/s, the equation simplifies to:
distance = 0.5 * acceleration * time^2
Plugging in the given values, we get:
distance = 0.5 * 1.4 m/s^2 * (14 s)^2 = 137.2 meters
2. Constant velocity:
After accelerating for 14 seconds, the bus travels at a constant velocity for 23 seconds. Since the velocity remains constant, the distance traveled during this phase can be calculated using the equation:
distance = velocity * time
We need to find the velocity of the bus during this phase. We know that acceleration is the rate of change of velocity, so integrating the acceleration over the 14-second period will give us the change in velocity. Using the equation:
velocity = initial velocity + (acceleration * time)
Since the initial velocity is 0 m/s and the acceleration is 1.4 m/s^2, plugging in the values, we get:
velocity = 0 m/s + (1.4 m/s^2 * 14 s) = 19.6 m/s
Now we can calculate the distance:
distance = 19.6 m/s * 23 s = 450.8 meters
3. Final deceleration:
After traveling at a constant speed for 23 seconds, the bus decelerates to a stop with an acceleration of magnitude 1.3 m/s^2. We can use the same equation as before to calculate the distance traveled during this phase. The final velocity will be 0 m/s, and the time is not given. Let's assume it takes t seconds for the bus to come to a stop.
0 m/s = 19.6 m/s + (-1.3 m/s^2 * t)
Solving for t, we find:
t = 15.077 seconds (approximated to three decimal places)
Now we can calculate the distance using:
distance = (velocity * time) + (0.5 * acceleration * time^2)
Plugging in the values:
distance = (19.6 m/s * 15.077 s) + (0.5 * -1.3 m/s^2 * (15.077 s)^2) = -140.6 meters
The negative sign indicates that the bus moved in the opposite direction during deceleration. Therefore, we take the magnitude of the distance to get:
distance = 140.6 meters
To find the total distance traveled by the bus, we sum up the distances from each phase:
total distance = 137.2 meters + 450.8 meters + 140.6 meters = 728.6 meters
Therefore, the total distance traveled by the bus is approximately 728.6 meters.