The velocity function (in meters per second) is given for a particle moving along a line. v (t) = 5t - 9, 0 ≤ t ≤ 3 (a) Find the displacement. - 9/2 Correct: Your answer is correct. m (b) Find the distance traveled by the particle during the given time interval.

a) The displacement is - 4.5 m.

b) The traveled distance is 11.7 m.

Explanation:

Hi there!

a) The velocity of the particle is the derivative of the displacement function, x (t):

v (t) = dx/dt = 5t - 9

Separating varibles:

dx = (5t - 9) dt

Integrating both sides from x = x0 to x and from t = 0 to t.

x - x0 = 1/2 · 5t² - 9t

x = 1/2 · 5t² - 9t + x0

If we place the origin of the system of reference at x = x0, the displacement at t = 3 will be x (3):

x (3) = 1/2 · 5 · (3) ² - 9 (3) + 0

x (3) = - 4.5

The displacement at t = 3 s is - 4.5 m. It means that the particle is located 4.5 m to the left from the origin of the system of reference at t = 3 s.

b) When the velocity is negative, the particle moves to the left. Let's find the time at which the velocity is less than zero:

v = 5t - 9

0 > 5t - 9

9/5 > t

1.8 s > t

Then until t = 1.8 s, the particle moves to the left from the origin of the reference system.

Let's find the position of the particle at that time:

x = 1/2 · 5t² - 9t

x = 1/2 · 5 (1.8 s) ² - 9 (1.8 s)

x = - 8.1 m

From t = 0 to t = 1.8 s the traveled distance is 8.1 m. After 1.8 s, the particle has positive velocity. It means that the particle is moving to the right, towards the origin. If at t = 3 the position of the particle is - 4.5 m, then the traveled distance from x = - 8.1 m to x = - 4.5 m is (8.1 m - 4.5 m) 3.6 m.

Then, the total traveled distance is (8.1 m + 3.6 m) 11.7 m.