A speedboat starts from rest and accelerates at +1.68 m/s2 for 5.43 s. At the end of this time, the boat continues for an additional 8.16 s with an acceleration of +0.475 m/s2. Following this, the boat accelerates at -0.866 m/s2 for 7.54 s. (a) What is the velocity of the boat at t = 21.13 s? (b) Find the total displacement of the boat.
To solve this problem, we can use the kinematic equations of motion, specifically the formula for final velocity and displacement.
(a) To find the velocity of the boat at t = 21.13 s, we need to calculate the velocities during each time interval and sum them up.
First, we calculate the velocity during the first time interval of 5.43 seconds.
Using the equation:
v = u + at,
where v is the final velocity, u is the initial velocity (which is 0 in this case), a is the acceleration, and t is the time.
v1 = 0 + (1.68 m/s^2) * 5.43 s
v1 = 9.12144 m/s (rounded to five decimal places)
Next, we calculate the velocity during the second time interval of 8.16 seconds.
v2 = v1 + (0.475 m/s^2) * 8.16 s
v2 = 13.5224 m/s (rounded to five decimal places)
Finally, we calculate the velocity during the third time interval of 7.54 seconds.
v3 = v2 + (-0.866 m/s^2) * 7.54 s
v3 = 7.56276 m/s (rounded to five decimal places)
Therefore, the velocity of the boat at t = 21.13 s is 7.56276 m/s (rounded to five decimal places).
(b) To find the total displacement of the boat, we need to calculate the displacements during each time interval and sum them up.
For the first time interval, we can use the formula:
s1 = u * t + 0.5 * a * t^2,
where s1 is the displacement, u is the initial velocity, a is the acceleration, and t is the time.
Using u = 0 and a = 1.68 m/s^2:
s1 = 0 * 5.43 s + 0.5 * 1.68 m/s^2 * (5.43 s)^2
s1 = 31.9586648 m (rounded to five decimal places)
For the second time interval, the initial velocity is v1 and the acceleration is 0.475 m/s^2:
s2 = v1 * t + 0.5 * a * t^2
s2 = 9.12144 m/s * 8.16 s + 0.5 * 0.475 m/s^2 * (8.16 s)^2
s2 = 138.71488 m (rounded to five decimal places)
For the third time interval, the initial velocity is v2 and the acceleration is -0.866 m/s^2:
s3 = v2 * t + 0.5 * a * t^2
s3 = 13.5224 m/s * 7.54 s + 0.5 * -0.866 m/s^2 * (7.54 s)^2
s3 = 69.91489 m (rounded to five decimal places)
The total displacement of the boat is the sum of all the individual displacements:
Total displacement = s1 + s2 + s3
Total displacement = 31.95865 m + 138.71488 m + 69.91489 m
Total displacement = 240.58842 m (rounded to five decimal places)
Therefore, the total displacement of the boat is 240.58842 m (rounded to five decimal places).