Questions: Below is the velocity function, in feet per second, for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. v(t) = t^3 - 14t^2 + 59t - 70 2 ≤ t ≤ 7 (a) Displacement: -30.8333 (b) Total distance: 35.1667

Below is the velocity function, in feet per second, for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval.
v(t) = t^3 - 14t^2 + 59t - 70 2 ≤ t ≤ 7
(a) Displacement: -30.8333
(b) Total distance: 35.1667

Transcript text: Below is the velocity function, in feet per second, for a particle moving along a straight line. Find (a) the displacement and (b) the total distance that the particle travels over the given interval. \[ v(t)=t^{3}-14 t^{2}+59 t-70 \quad 2 \leq t \leq 7 \] (a) Displacement: -30.8333 (b) Total distance: 35.1667

Solution

Solution Steps

To solve this problem, we need to integrate the velocity function over the given interval to find both the displacement and the total distance traveled by the particle.

(a) Displacement: This is found by calculating the definite integral of the velocity function \( v(t) \) from \( t = 2 \) to \( t = 7 \).

(b) Total Distance: This is found by integrating the absolute value of the velocity function over the same interval, which accounts for any changes in direction.

Step 1: Define the Problem

We are given a velocity function \( v(t) = t^3 - 14t^2 + 59t - 70 \) and need to find the displacement and total distance traveled by a particle over the interval \( 2 \leq t \leq 7 \).

Step 2: Calculate Displacement

The displacement is the definite integral of the velocity function over the interval \([2, 7]\). This is given by: \[ \text{Displacement} = \int_{2}^{7} (t^3 - 14t^2 + 59t - 70) \, dt \] Evaluating this integral, we find: \[ \text{Displacement} = -30.8333 \]

Step 3: Calculate Total Distance

The total distance is the integral of the absolute value of the velocity function over the same interval. This accounts for any changes in direction: \[ \text{Total Distance} = \int_{2}^{7} |t^3 - 14t^2 + 59t - 70| \, dt \] Evaluating this integral, we find: \[ \text{Total Distance} = 35.1667 \]