A speedboat starts from rest and accelerates at +1.94 m/s2 for 7.00 s. At the end of this time, the boat continues for an additional 6.00 s with an acceleration of +0.515 m/s2. Following this, the boat accelerates at -1.58 m/s2 for 8.00 s. What is the velocity of the boat at t = 21.0 s?

Question

A speedboat starts from rest and accelerates at +1.94 m/s2 for 7.00 s. At the end of this time, the boat continues for an additional 6.00 s with an acceleration of +0.515 m/s2. Following this, the boat accelerates at -1.58 m/s2 for 8.00 s. What is the velocity of the boat at t = 21.0 s?

Answer 1

To find the velocity of the boat at t = 21.0 s, we need to calculate the total displacement of the boat at this time. We can then use this displacement along with the given time interval to calculate the velocity using the formula:

v = u + at

where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time interval.

Let's break down the given information into separate time intervals and calculate the displacements and velocities for each interval.

First, from t = 0 to t = 7.00 s:
Initial velocity (u1) = 0 (as the boat starts from rest)
Acceleration (a1) = +1.94 m/s^2
Time interval (t1) = 7.00 s

Using the formula for displacement:
s1 = u1*t1 + (1/2)*a1*t1^2
s1 = 0 + (1/2)*1.94*(7.00)^2

Next, from t = 7.00 s to t = 13.00 s:
Initial velocity (u2) = velocity at the end of the first interval
Acceleration (a2) = +0.515 m/s^2
Time interval (t2) = 6.00 s

Using the formula for displacement:
s2 = u2*t2 + (1/2)*a2*t2^2
s2 = s1 + u2 * t2 + (1/2)*a2*(t2)^2

Finally, from t = 13.00 s to t = 21.00 s:
Initial velocity (u3) = velocity at the end of the second interval
Acceleration (a3) = -1.58 m/s^2
Time interval (t3) = 8.00 s

Using the formula for displacement:
s3 = u3*t3 + (1/2)*a3*t3^2
s3 = s2 + u3 * t3 + (1/2)*a3*(t3)^2

Now let's calculate the displacements and velocities.

For the first interval:
s1 = 0 + (1/2)*1.94*(7.00)^2

For the second interval:
s2 = s1 + u2 * t2 + (1/2)*a2*(t2)^2

For the third interval:
s3 = s2 + u3 * t3 + (1/2)*a3*(t3)^2

To find the total displacement at t = 21.0 s, we add up the displacements of all three intervals.
Total displacement = s1 + s2 + s3

Then, we can calculate the velocity at t = 21.0 s using the formula:
v = u + at

where:
v is the final velocity,
u is the initial velocity (which is the velocity at the end of the third interval),
a is the acceleration (which is the acceleration during the third interval), and
t is the time interval (which is 21.0 s).

By plugging in the values into the formula, we can find the velocity of the boat at t = 21.0 s.