A sailboat starts from rest and accelerates at a rate of 0.12 m/s 2 over a distance of 280 m.

(a) Find the magnitude of the boat's final velocity. m/s
(b) Find the time it takes the boat to travel this distance.

v=sqrt(2as) = sqrt(2•0.12•280)=8.2 m/s.

s=at²/2,
t=sqrt{2s/a}=sqrt(2•280/0.12) = 68.3 s.

(a) Well, if the sailboat starts from rest and accelerates at a rate of 0.12 m/s^2, it's like watching a very slow-motion sailboat race. So, to find the magnitude of the boat's final velocity, we can use the equation:

v = u + at,

Where:
v = final velocity,
u = initial velocity (which is zero in this case),
a = acceleration,
and t = time.

Since the boat starts from rest, u = 0. Plugging in the values, we get:

v = 0 + (0.12 m/s^2)(280 m).

Calculating that gives us the final velocity of the sailboat.

(b) Now, to find the time it takes for the boat to travel this distance, we can use another equation:

s = ut + (1/2)at^2,

Where:
s = distance,
u = initial velocity (which is still zero),
a = acceleration,
and t = time.

Rearranging this equation to solve for t, we get:

t = sqrt(2s / a).

Plugging in the values for s and a, we can calculate the time it takes for the sailboat to travel this distance.

Now, let me grab my calculator and put on my sailor hat!

To solve this problem, we can use the kinematic equation:

vf^2 = vi^2 + 2ad

where:
- vf is the final velocity
- vi is the initial velocity (which is assumed to be zero in this case)
- a is the acceleration
- d is the distance traveled

(a) Magnitude of the boat's final velocity (vf):
Using the equation, we can solve for vf:

vf^2 = 0^2 + 2 * 0.12 m/s^2 * 280 m
vf^2 = 0 + 67.2 m^2/s^2
vf = sqrt(67.2) m/s
vf ≈ 8.20 m/s

Therefore, the magnitude of the boat's final velocity is approximately 8.20 m/s.

(b) Time it takes the boat to travel the distance (t):
To find the time, we can use another kinematic equation:

vf = vi + at

Rearranging the equation, we have:

t = (vf - vi) / a
t = (8.20 m/s - 0 m/s) / 0.12 m/s^2
t = 8.20 m/s / 0.12 m/s^2
t ≈ 68.33 s

Therefore, it takes approximately 68.33 seconds for the boat to travel this distance.

To find the answers, we can use the equations of motion. The equations relevant to this problem are:

1. Final velocity equation: v = u + at
2. Distance equation: s = ut + (1/2)at^2

where:
- v is the final velocity
- u is the initial velocity (which is 0 in this case, since the sailboat starts from rest)
- a is the acceleration (0.12 m/s^2)
- t is the time taken
- s is the distance (280 m)

(a) To find the magnitude of the boat's final velocity:
Using equation (1), we substitute the given values:

v = u + at
v = 0 + (0.12 m/s^2)(t)

Since the boat starts from rest, the initial velocity (u) is 0. Therefore, the equation becomes:

v = 0.12 m/s^2 * t

Now, we need to find the value of t. We can rearrange equation (2) to solve for t:

s = ut + (1/2)at^2
280 m = 0 + (1/2)(0.12 m/s^2)t^2

Rearranging this equation to solve for t, we get:

t^2 = (2 * 280 m) / (0.12 m/s^2)
t^2 = 4666.67

Now, we take the square root of both sides to find the value of t:

t ≈ 68.25 s (approximately)

Substituting the value of t back into the equation for v:

v = 0.12 m/s^2 * 68.25 s
v ≈ 8.19 m/s

Therefore, the magnitude of the boat's final velocity is approximately 8.19 m/s.

(b) To find the time it takes the boat to travel this distance:
We have already found the value of t above, which is approximately 68.25 s. Therefore, the time it takes the boat to travel this distance is approximately 68.25 seconds.