A little bumblebee accelerates uniformly at 1.5 m/s2 from rest to 22 m/s. Immediately upon reaching 22m/s, the brakes are applied and it stops 2.5 s later. Find the total distance traveled.
Phase 1
v = a t
22 = 1.5 t
t = 22/1.5
average speed = 11
so x = 11 *22/1.5
Phase 2
starts at x = 11*22/1.5 and t = 22/1.5
average speed during stop = 11 again
time to stop = 2.5
so stopping distance = 11 * 2.5
so
x = 11*22/1.5 + 11*2.5
To find the total distance traveled, we need to find the distances traveled during acceleration and deceleration separately and then add them together.
Let's start with the acceleration phase. We can use the formula:
v = u + at
Where:
v = final velocity (22 m/s)
u = initial velocity (0 m/s, as it starts from rest)
a = acceleration (1.5 m/s²)
t = time taken
Rearranging the formula to find the time taken during acceleration:
t = (v - u) / a
Substituting the values, we find:
t = (22 m/s - 0 m/s) / 1.5 m/s²
= 22 m/s / 1.5 m/s²
≈ 14.67 s
The time taken during acceleration is approximately 14.67 seconds.
Now, let's calculate the distance traveled during acceleration using the formula:
s = ut + (1/2)at²
Substituting the values:
s = 0 m/s * 14.67 s + (1/2)(1.5 m/s²)(14.67 s)²
= 0 m + (1/2)(1.5 m/s²)(14.67 s)(14.67 s)
≈ 153.9 m
The distance traveled during acceleration is approximately 153.9 meters.
Next, let's calculate the deceleration phase.
The bumblebee stops in 2.5 seconds. So, the time taken during deceleration is 2.5 seconds.
Using the formula v = u + at, we can find the acceleration during deceleration.
Final velocity, v = 0 m/s (as the bumblebee stops)
Initial velocity, u = 22 m/s
Time taken, t = 2.5 s
0 = 22 m/s + a * 2.5 s
Rearranging the formula:
a = -22 m/s / 2.5 s
= -8.8 m/s²
The acceleration during deceleration is -8.8 m/s².
Now, let's calculate the distance traveled during deceleration:
Using the formula s = ut + (1/2)at²:
s = 22 m/s * 2.5 s + (1/2)(-8.8 m/s²)(2.5 s)²
= 55 m + (1/2)(-8.8 m/s²)(2.5 s)(2.5 s)
= 55 m - (1/2)(8.8 m/s²)(6.25 s²)
= 55 m - (4.4 m/s²)(6.25 s²)
= 55 m - 27.5 m
= 27.5 m
The distance traveled during deceleration is 27.5 meters.
Finally, to find the total distance traveled, we add the distances traveled during acceleration and deceleration:
Total distance traveled = distance during acceleration + distance during deceleration
= 153.9 m + 27.5 m
≈ 181.4 m
Therefore, the total distance traveled by the bumblebee is approximately 181.4 meters.
To find the total distance traveled by the bumblebee, we need to break down the motion into two parts: the acceleration phase and the deceleration phase.
First, let's calculate the distance traveled during the acceleration phase. We can use the equation:
v = u + at,
where:
v = final velocity = 22 m/s,
u = initial velocity = 0 m/s (as the bumblebee starts from rest), and
a = acceleration = 1.5 m/s^2.
Rearranging the equation, we have:
t = (v - u) / a.
Substituting the given values, we get:
t = (22 m/s - 0 m/s) / 1.5 m/s^2 = 14.67 s.
Now, we can calculate the distance traveled during the acceleration phase using the equation:
s = ut + (1/2)at^2.
Since the initial velocity (u) is 0, the equation simplifies to:
s = (1/2)at^2.
Plugging in the values, we have:
s = (1/2) * 1.5 m/s^2 * (14.67 s)^2 = 157.79 m.
Next, let's calculate the distance traveled during the deceleration phase. We know that the bumblebee stops 2.5 s after reaching 22 m/s. During this time, the bumblebee will continue to move with the same magnitude of velocity (22 m/s) but in the opposite direction. Therefore, the deceleration (a2) can be calculated as:
a2 = v2 / t2,
where:
v2 = final velocity = 22 m/s (opposite direction) and
t2 = time taken to stop = 2.5 s.
Substituting the given values:
a2 = (22 m/s) / 2.5 s = 8.8 m/s^2.
Now, we can calculate the distance traveled during the deceleration phase using the equation:
s2 = v2t2 - (1/2)a2t2^2.
Plugging in the values, we get:
s2 = (22 m/s)(2.5 s) - (1/2)(8.8 m/s^2)(2.5 s)^2 = 27.5 m - 27.5 m = 0 m.
Since the bumblebee comes to a complete stop during the deceleration phase, the distance traveled is 0 m.
Finally, we can calculate the total distance traveled by adding the distances traveled during the acceleration and deceleration phases:
Total distance = s + s2 = 157.79 m + 0 m = 157.79 m.
Therefore, the total distance traveled by the bumblebee is 157.79 meters.