The displacement (in meters) of a particle moving in a straight line is given by s = 4 t^3 where t is measured in seconds. Find the average velocity of the particle over the time interval [ 7 , 9].
Find the (instantaneous) velocity of the particle when t = 7.
Why did the particle bring a calculator to the party?
Because it wanted to calculate its average velocity and instantaneous velocity, of course!
To find the average velocity of the particle over the time interval [7, 9], we can use the formula:
Average Velocity = (Change in Displacement) / (Change in Time)
Given that the displacement function is s = 4t^3, we can plug in the values to calculate the change in displacement:
Change in Displacement = s(9) - s(7)
= (4(9^3)) - (4(7^3))
= 4(729) - 4(343)
= 2916 - 1372
= 1544 meters
Now, let's calculate the change in time:
Change in Time = 9 - 7
= 2 seconds
Finally, we can find the average velocity:
Average Velocity = (Change in Displacement) / (Change in Time)
= 1544 / 2
= 772 meters/second
Now, for the instantaneous velocity when t = 7. We can find this by taking the derivative of the displacement function with respect to time:
s(t) = 4t^3
Taking the derivative, we get:
v(t) = d/dt (4t^3)
= 12t^2
Plugging in t = 7, we can calculate the instantaneous velocity:
v(7) = 12(7^2)
= 12(49)
= 588 meters/second
So, the instantaneous velocity when t = 7 is 588 meters/second.
Hope that put a smile on your face!
To find the average velocity of the particle over the time interval [7, 9], we need to calculate the change in displacement divided by the change in time.
The change in displacement is given by the difference in the values of s at the end and the beginning of the time interval. Let's calculate the displacement at t = 7 and t = 9.
Substituting t = 7 into the displacement equation, we have:
s(7) = 4 * (7)^3 = 4 * 343 = 1372 meters
Substituting t = 9 into the displacement equation, we have:
s(9) = 4 * (9)^3 = 4 * 729 = 2916 meters
Now, we can calculate the change in displacement:
Δs = s(9) - s(7) = 2916 - 1372 = 1544 meters
The change in time is the difference in the values of t at the end and the beginning of the time interval. In this case, the time interval is [7, 9], so Δt = 9 - 7 = 2 seconds.
Finally, we can calculate the average velocity:
Average velocity = Δs / Δt = 1544 / 2 = 772 meters/second
To find the instantaneous velocity of the particle when t = 7, we need to calculate the derivative of the displacement equation with respect to time and evaluate it at t = 7.
Taking the derivative of s = 4t^3:
v(t) = d s / d t = 12t^2
To find the instantaneous velocity at t = 7, substitute t = 7 into the velocity equation:
v(7) = 12(7)^2 = 12 * 49 = 588 meters/second
Therefore, the average velocity of the particle over the time interval [7, 9] is 772 meters/second, and the instantaneous velocity of the particle when t = 7 is 588 meters/second.