Ive tried numerous methods, but I cant seem to get this right. Please help, and explain step by step.
An object's position is given by x=bt+ct^3, where b=1.50m/s and c=0.640m/s3. To study the limiting process leading to the instantaneous velocity, calculate the object's average velocity over time intervals from 1.00 s to 3.00 s.
Also, from 1.95 s to 2.05 s.
well, just for fun, let's see hat the form of the limit is
v = dx/dt = b + 3 c t^2
= 1.5 + 1.92 t^2
=========================
now
from 1 to 3
x(3)= 1.5*3 +.64*27=4.5+17.28
= 21.78
x(1) = 1.5+.64 = 2.14
so
x(3)-x(1) = 19.64
delta t = 3 - 1 = 2 seconds
so
v during interval = 19.64/2
= 9.82 m/s
now check that with calculus v in the middle of that range
v(2) =1.5+ 1.92 (4) = 9.18 m/s
that makes sense, the parabola is headed up so the instant speed i the middle should be a bit less than average.
Do you see what is going on? try some other intervals.
Also, from 1.95 s to 2.05 s
this is set up so you can compare your calculated speed with the real speed at 2 seconds from calculus. By coincidence we already did that last part, v(2) = 9.18 m/s and your average from 1.95 to 2.05 should be very close to that, like really, really close ;)
Thank you!
To calculate the object's average velocity over a given time interval, we need to find the change in position (Δx) divided by the change in time (Δt). Here's how we can calculate the average velocity for each interval:
1) Time interval from 1.00 s to 3.00 s:
Step 1: Calculate the position at the starting time (x1) and the ending time (x2):
For t = 1.00 s:
x1 = b(1.00) + c(1.00)^3
For t = 3.00 s:
x2 = b(3.00) + c(3.00)^3
Step 2: Calculate the change in position (Δx) and change in time (Δt):
Δx = x2 - x1
Δt = 3.00 s - 1.00 s
Step 3: Calculate the average velocity:
Average velocity = Δx / Δt
2) Time interval from 1.95 s to 2.05 s:
Step 1: Calculate the position at the starting time (x1) and the ending time (x2):
For t = 1.95 s:
x1 = b(1.95) + c(1.95)^3
For t = 2.05 s:
x2 = b(2.05) + c(2.05)^3
Step 2: Calculate the change in position (Δx) and change in time (Δt):
Δx = x2 - x1
Δt = 2.05 s - 1.95 s
Step 3: Calculate the average velocity:
Average velocity = Δx / Δt
Now, let's go ahead and calculate the average velocity for each interval:
1) Time interval from 1.00 s to 3.00 s:
- Substitute the values of x1 and x2 into the equations:
x1 = 1.50(1.00) + 0.640(1.00)^3
x2 = 1.50(3.00) + 0.640(3.00)^3
- Calculate Δx and Δt:
Δx = x2 - x1
Δt = 3.00 s - 1.00 s
- Substitute the values of Δx and Δt into the equation for average velocity:
Average velocity = Δx / Δt
2) Time interval from 1.95 s to 2.05 s:
- Substitute the values of x1 and x2 into the equations:
x1 = 1.50(1.95) + 0.640(1.95)^3
x2 = 1.50(2.05) + 0.640(2.05)^3
- Calculate Δx and Δt:
Δx = x2 - x1
Δt = 2.05 s - 1.95 s
- Substitute the values of Δx and Δt into the equation for average velocity:
Average velocity = Δx / Δt
By following these steps, you should be able to calculate the average velocity for both time intervals.