A sailboat starts from rest and accelerates at a rate of 0.18 m/s2 over a distance of 454 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.
s
v =sqrt (2as)
t=v/a
Can you help me solve this problem?
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To find the magnitude of the boat's final velocity, we can use the formula:
v^2 = u^2 + 2as,
where v is the final velocity, u is the initial velocity (which is 0 since the boat starts from rest), a is the acceleration, and s is the distance traveled.
(a) To calculate the magnitude of the boat's final velocity, we can substitute the given values into the formula:
v^2 = 0^2 + 2(0.18 m/s^2)(454 m).
Simplifying the equation gives:
v^2 = 0 + 2(0.18 m/s^2)(454 m),
v^2 = 0 + 2(81.72 m^2/s^2),
v^2 = 163.44 m^2/s^2.
To find v, we take the square root of both sides:
v = √163.44 m^2/s^2,
v ≈ 12.80 m/s.
Therefore, the magnitude of the boat's final velocity is approximately 12.80 m/s.
(b) To find the time it takes the boat to travel this distance, we can use the equation:
s = ut + (1/2)at^2,
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time taken.
Since the boat starts from rest, the initial velocity u is 0. We can simplify the equation to:
s = (1/2)at^2,
454 m = (1/2)(0.18 m/s^2)t^2.
To solve for t, we can rearrange the equation:
t^2 = (2 * 454 m) / 0.18 m/s^2,
t^2 = 2,527.78 s^2.
Taking the square root of both sides gives:
t = √2,527.78 s^2,
t ≈ 50.28 s.
Therefore, it takes approximately 50.28 seconds for the boat to travel a distance of 454 meters.