t(sec) | 0 | 2 | 4 | 6 | 8 |
____________|_____|_____|_____|____|_____|
a(t) (ft/sec²) | 2 | 3 | 4 | 3 | 2 |
The table for the acceleration of a particle from 0 to 8 seconds is given in the table above. If the velocity at t=0 is 4 feet per second, the approximation value of the velocity, in feet per second, at t=8 seconds, computed using the Riemann Sum with four subdivisions of equal length is...
To approximate the velocity at t=8 seconds using the Riemann Sum with four subdivisions of equal length, we need to find the area under the acceleration curve over the interval from 0 to 8 seconds.
First, let's find the time interval for each subdivision:
8 seconds / 4 subdivisions = 2 seconds per subdivision.
Next, let's calculate the velocity at each subdivision by finding the area under the acceleration curve for each subdivision.
For the first subdivision (0 to 2 seconds):
Acceleration = 2 ft/sec²
Time interval = 2 seconds
Area = Acceleration * Time interval = 2 ft/sec² * 2 seconds = 4 ft/sec
For the second subdivision (2 to 4 seconds):
Acceleration = 3 ft/sec²
Time interval = 2 seconds
Area = Acceleration * Time interval = 3 ft/sec² * 2 seconds = 6 ft/sec
For the third subdivision (4 to 6 seconds):
Acceleration = 4 ft/sec²
Time interval = 2 seconds
Area = Acceleration * Time interval = 4 ft/sec² * 2 seconds = 8 ft/sec
For the fourth subdivision (6 to 8 seconds):
Acceleration = 3 ft/sec²
Time interval = 2 seconds
Area = Acceleration * Time interval = 3 ft/sec² * 2 seconds = 6 ft/sec
Finally, let's sum up the areas for all four subdivisions to approximate the total change in velocity:
Total change in velocity = 4 ft/sec + 6 ft/sec + 8 ft/sec + 6 ft/sec = 24 ft/sec.
To find the approximate value of the velocity at t=8 seconds, we add the total change in velocity to the initial velocity at t=0:
Velocity at t=8 seconds = Initial velocity + Total change in velocity = 4 ft/sec + 24 ft/sec = 28 ft/sec.
Therefore, the approximation value of the velocity at t=8 seconds, computed using the Riemann Sum with four subdivisions of equal length, is 28 feet per second.
To approximate the value of velocity at t=8 seconds using the Riemann Sum with four subdivisions of equal length, we need to first calculate the change in time (Δt) and then compute the Riemann Sum using the given data.
Calculating Δt:
Since we have 4 subdivisions of equal length, the total time interval (8 seconds - 0 seconds) is divided into 4 equal subintervals. Therefore, each subinterval will have a length of Δt = (8 seconds - 0 seconds) / 4 = 2 seconds.
Now, let's compute the Riemann Sum to approximate the velocity at t=8 seconds.
Riemann Sum:
The Riemann Sum is calculated by summing up the products of the average acceleration in each subinterval and the corresponding change in time.
First, let's calculate the average acceleration for each subinterval:
For the first subinterval (0 to 2 seconds):
Average acceleration = (2 + 3) / 2 = 2.5 ft/sec²
For the second subinterval (2 to 4 seconds):
Average acceleration = (3 + 4) / 2 = 3.5 ft/sec²
For the third subinterval (4 to 6 seconds):
Average acceleration = (4 + 3) / 2 = 3.5 ft/sec²
For the fourth subinterval (6 to 8 seconds):
Average acceleration = (3 + 2) / 2 = 2.5 ft/sec²
Next, let's calculate the Riemann Sum:
Riemann Sum = (Average acceleration in the first subinterval * Δt) + (Average acceleration in the second subinterval * Δt) + (Average acceleration in the third subinterval * Δt) + (Average acceleration in the fourth subinterval * Δt)
Riemann Sum = (2.5 ft/sec² * 2 seconds) + (3.5 ft/sec² * 2 seconds) + (3.5 ft/sec² * 2 seconds) + (2.5 ft/sec² * 2 seconds)
Riemann Sum = 5 ft/sec + 7 ft/sec + 7 ft/sec + 5 ft/sec
Riemann Sum = 24 ft/sec
Therefore, the approximation value of the velocity at t=8 seconds, computed using the Riemann Sum with four subdivisions of equal length, is 24 feet per second.