The acceleration of a particle along a straight line is defined by a = (2t - 9) m/s2, where t is in seconds. At t = 0, s = 1 m and v = 10 m/s. When t = 9 s, determine (a) the particle’s position, (b) the total distance traveled, and (c) the velocity.
To determine the particle's position, total distance traveled, and velocity at t = 9 s, we can integrate the given acceleration function over the time interval 0 to 9 seconds.
(a) Particle's position:
To find the particle's position, we need to integrate the acceleration function with respect to time twice.
First, integrate the given acceleration function (a = 2t - 9) with respect to time to obtain the velocity function:
∫(2t - 9) dt = t^2 - 9t + C1
Since the initial velocity is given as 10 m/s, we can substitute t = 0 and v = 10 into the velocity equation to solve for the constant C1:
0^2 - 9(0) + C1 = 10
C1 = 10
Therefore, the velocity function is:
v = t^2 - 9t + 10
Next, integrate the velocity function with respect to time to obtain the position function:
∫(t^2 - 9t + 10) dt = (1/3)t^3 - (9/2)t^2 + 10t + C2
Since the initial position is given as 1 m, we can substitute t = 0 and s = 1 into the position equation to solve for the constant C2:
(1/3)(0)^3 - (9/2)(0)^2 + 10(0) + C2 = 1
C2 = 1
Therefore, the position function is:
s = (1/3)t^3 - (9/2)t^2 + 10t + 1
To find the particle's position at t = 9 s, substitute t = 9 into the position function:
s = (1/3)(9)^3 - (9/2)(9)^2 + 10(9) + 1
s ≈ 19.5 m
So, at t = 9 s, the particle's position is approximately 19.5 m.
(b) Total distance traveled:
To find the total distance traveled, integrate the absolute value of the velocity function from t = 0 to t = 9:
∫|t^2 - 9t + 10| dt
To calculate this integral, divide the integration interval into two parts: from t = 0 to t = 3 and from t = 3 to t = 9. In both intervals, the velocity is positive, so we don't have to take the absolute value. Integrate each part separately:
∫(t^2 - 9t + 10) dt from t = 0 to t = 3
and
∫(t^2 - 9t + 10) dt from t = 3 to t = 9
Calculate the integrals using the earlier expression for the velocity function, and add the absolute values of the results. The sum will give you the total distance traveled.
(c) Velocity:
To find the velocity at t = 9 s, substitute t = 9 into the velocity function:
v = (9)^2 - 9(9) + 10
v = 19 m/s
So, at t = 9 s, the velocity of the particle is 19 m/s.