A sailboat starts from rest and accelerates at a rate of 0.12 m/s2 over a distance of 280 m.
(a) Find the magnitude of the boat's final velocity.
(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.
To find the magnitude of the boat's final velocity, we can use the equation:
v_f = sqrt(v_i^2 + 2 * a * d)
Where:
v_f = final velocity
v_i = initial velocity (which is 0 in this case)
a = acceleration
d = distance
(a) Let's plug in the values into the equation:
v_f = sqrt(0^2 + 2 * 0.12 * 280)
Simplifying this equation:
v_f = sqrt(0 + 67.2)
v_f ≈ sqrt(67.2)
Using a calculator, we find that:
v_f ≈ 8.2 m/s
So, the magnitude of the boat's final velocity is approximately 8.2 m/s.
(b) To find the time it takes for the boat to travel this distance, we can use the equation:
d = (v_i * t) + (0.5 * a * t^2)
In this case, the initial velocity (v_i) is 0, so the equation simplifies to:
d = (0.5 * a * t^2)
Let's rearrange the equation and solve for time (t):
t^2 = (2 * d) / a
t = sqrt((2 * d) / a)
Now we can plug in the values:
t = sqrt((2 * 280) / 0.12)
Calculating this equation:
t = sqrt(5600 / 0.12)
t ≈ sqrt(46666.67)
Using a calculator, we find that:
t ≈ 215.8 seconds
So, it takes approximately 215.8 seconds for the boat to travel this distance.
To find the magnitude of the boat's final velocity, you can use the following equation:
v^2 = u^2 + 2as
Where:
v = final velocity
u = initial velocity (which is 0 since the boat starts from rest)
a = acceleration
s = distance traveled
Substituting the given values into the equation, we have:
v^2 = 0^2 + 2(0.12)(280) = 0 + 67.2 = 67.2
To find the magnitude of the final velocity, we take the square root of 67.2:
v = √67.2 ≈ 8.2 m/s
So the magnitude of the boat's final velocity is approximately 8.2 m/s.
To find the time it takes the boat to travel the distance, we can use another equation:
s = ut + (1/2)at^2
Where:
s = distance traveled (280 m in this case)
u = initial velocity (0 m/s since the boat starts from rest)
a = acceleration (0.12 m/s^2)
t = time
Substituting the values into the equation, we have:
280 = 0(0) + (1/2)(0.12)t^2
280 = 0 + 0.06t^2
280/0.06 = t^2
t^2 = 4666.67
To find t, we take the square root of 4666.67:
t = √4666.67 ≈ 68.28 s
So it takes approximately 68.28 seconds for the boat to travel the distance of 280 m.