the velocity of a particle traveling in a straight line is given by v=(6t-3t^2)m/s, where t is in seconds, if s=0 when t= 0. determine the particles deceleration and position when t=3s. how far has the particle traveled during the 3s time interval and what is its average speed?
Thanks. Very cool.
To determine the particle's deceleration and position when t = 3s, we need to find the expression for acceleration and integrate it to find the position function.
Step 1: Find the acceleration function:
Given velocity function, v = (6t - 3t^2) m/s.
Acceleration, a, is the derivative of velocity with respect to time, t:
a = dv/dt = d/dt(6t - 3t^2) = 6 - 6t
Step 2: Determine the deceleration when t = 3s:
Substitute t = 3 into the acceleration function:
a = 6 - 6(3) = 6 - 18 = -12 m/s^2.
The deceleration is -12 m/s^2.
Step 3: Integrate the velocity function to find the position function:
Position, s, is the integral of velocity with respect to time, t:
s = ∫v dt = ∫(6t - 3t^2) dt = 3t^2 - t^3 + C
Given that s = 0 when t = 0, we can find the constant C:
0 = 3(0)^2 - (0)^3 + C
0 = 0 + C
C = 0
So, the position function is s = 3t^2 - t^3.
Step 4: Determine the position when t = 3s:
Substitute t = 3 into the position function:
s = 3(3)^2 - (3)^3
s = 3(9) - 27
s = 27 - 27
s = 0
The particle's position when t = 3s is 0.
Step 5: Calculate the distance traveled during the 3s time interval:
To calculate the distance traveled, we need to consider the absolute value of the displacement.
The displacement can be found by subtracting the initial position (t = 0) from the final position (t = 3s):
Displacement = s(3) - s(0) = 0 - 0 = 0
Therefore, the distance traveled during the 3s time interval is 0 meters.
Step 6: Calculate the average speed:
Average speed is the total distance traveled divided by the time interval.
Average speed = Distance traveled / Time interval = 0m / 3s = 0 m/s.
In summary:
- The particle's deceleration when t = 3s is -12 m/s^2.
- The particle's position when t = 3s is 0 meters.
- The particle has traveled a distance of 0 meters during the 3s time interval.
- The average speed of the particle during the 3s time interval is 0 m/s.
To determine the particle's deceleration, we need to find its acceleration first. The acceleration can be found by taking the derivative of the velocity function with respect to time (t).
Given that the velocity function is v = (6t - 3t^2) m/s, we can differentiate it to find the acceleration function (a):
a = dv/dt = d(6t - 3t^2)/dt
= 6 - 6t
Now, to find the particle's deceleration, we need to substitute the given value of t = 3s into the acceleration function:
a(t=3) = 6 - 6(3)
= -12 m/s^2
So, the particle's deceleration is -12 m/s^2.
To find the position of the particle when t = 3s, we need to integrate the velocity function with respect to time t. The integral of v with respect to t will give us the position function, which we'll call s.
s = ∫(6t - 3t^2) dt
= 6(1/2)t^2 - 3(1/3)t^3 + C
= 3t^2 - t^3 + C
To find the constant of integration (C), we know that s = 0 when t = 0. Substituting these values, we can solve for C:
0 = 3(0)^2 - (0)^3 + C
C = 0
Therefore, the position of the particle when t = 3s is:
s(t=3) = 3(3)^2 - (3)^3
= 27 - 27
= 0 m
The particle has come back to its starting position when t = 3s.
The distance traveled during the 3s time interval can be found by calculating the total displacement. Since the particle has traveled back to its starting position, the displacement is zero. Therefore, the particle has not covered any distance during this time interval.
The average speed can be calculated by taking the total distance traveled (which is zero) and dividing it by the total time, which is 3s. So, the average speed is:
Average speed = (total distance)/(total time)
= 0/3
= 0 m/s
Therefore, the average speed of the particle during the 3s time interval is 0 m/s.
if v = 6t - 3t^2
then a = 6 - 6t
and s = distance = 3t^2 - t^3 + c , where c is a constant.
but we are told that s-0 when t= 0 , so c = 0
when t = 3
a = 6 - 6(3) = - -12
s = 3(3^2) - 3^3 = 0
total distance = final position - initial position
= 0 - 0 = 0
avg speed = 0/3 = 0