A speedboat starts from rest and accelerates at +1.99 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 -1.44 m/s2 for 8.00 s. What is the velocity of the boat at t = 21.0 s?
The velocity change during each interval is (acceleration rate)*(time)
Add the velocity changes for the three intervals. The velocity will drop durng the third interval because of deceleration (negative acceleration).
To find the velocity of the boat at t = 21.0 s, we need to consider the three different time intervals and calculate the change in velocity for each.
Step 1: Find the change in velocity during the first interval (0 s - 7.0 s).
Using the formula:
v = u + a * t
Where:
v = final velocity
u = initial velocity (which is 0 since the boat starts from rest)
a = acceleration (+1.99 m/s2)
t = time interval (7.0 s)
So, for the first interval:
v1 = 0 + (1.99 m/s2) * (7.0 s) = 13.93 m/s
Step 2: Find the change in velocity during the second interval (7.0 s - 13.0 s).
Using the same formula:
v = u + a * t
Where:
u = final velocity of the first interval (13.93 m/s)
a = acceleration (+0.515 m/s2)
t = time interval (6.0 s)
So, for the second interval:
v2 = 13.93 m/s + (0.515 m/s2) * (6.0 s) = 17.89 m/s
Step 3: Find the change in velocity during the third interval (13.0 s - 21.0 s).
Using the same formula:
v = u + a * t
Where:
u = final velocity of the second interval (17.89 m/s)
a = acceleration (-1.44 m/s2)
t = time interval (8.0 s)
So, for the third interval:
v3 = 17.89 m/s + (-1.44 m/s2) * (8.0 s) = 5.37 m/s
Step 4: Calculate the total velocity at t = 21.0 s.
To get the total velocity, we need to sum up the change in velocity during each interval.
So, the velocity at t = 21.0 s is:
v_total = v1 + v2 + v3
= 13.93 m/s + 17.89 m/s + 5.37 m/s
= 37.19 m/s
Therefore, the velocity of the boat at t = 21.0 s is 37.19 m/s.