The velocity of a particle moving along the 𝑥-axis is given by 𝑓(𝑡)=9−3𝑡 cm/sec. Use a graph of 𝑓(𝑡) to find the exact change in position of the particle from time 𝑡=0 to 𝑡=4 seconds.
if 𝑓(𝑡)=9−3𝑡 , represents the velocity, then the displacement function would be
s(t) = 9t - 3/2 t^2 + c
s(0) = 0-0 + c
s(4) = 36 - 3/2(16) + c
= 36 - 24 + c = 12+c
change in position = 12+c - c = 12 cm
To find the exact change in position of the particle from time t=0 to t=4 seconds, we need to integrate the velocity function f(t) with respect to t over the given time interval.
The function f(t) represents the velocity of the particle at any given time t along the x-axis. To integrate this function, we can use the definite integral, which calculates the area under the velocity curve between the two time points.
The integral of f(t) with respect to t over the interval [0, 4] gives us the change in position of the particle during this time period.
∫[0,4] (9 - 3t) dt
To evaluate this integral, we can use the power rule of integration. Integrating each term separately:
∫[0,4] 9 dt - ∫[0,4] 3t dt
The integral of a constant term, such as 9, with respect to t is simply the constant times t:
9t | [0,4] - ∫[0,4] 3t dt
Now, let's integrate the second term:
9t | [0,4] - 3∫[0,4] t dt
Applying the power rule of integration to the second term, we get:
9t | [0,4] - 3[(t^2)/2] | [0,4]
Plugging in the upper and lower limits of integration:
(9 * 4) - 3[(4^2)/2] - [(9 * 0) - 3[(0^2)/2]]
Simplifying further:
36 - 3[8/2] - 0
36 - 12 - 0
The exact change in position of the particle from t=0 to t=4 seconds is 24 cm.
Therefore, the particle moves 24 cm along the x-axis during this time interval.