Question

Find the total distance traveled by a particle according to the velocity function v(t)=−4t+6 m/sec over the time interval [0,5]. Enter your answer as an exact fraction if necessary and do not include units in your answer.

Answer 1

Answer: 42 m

Explanation: To find the total distance traveled by the particle over the time interval [0,5], we need to find the area under the velocity function v(t) over the interval. We can use the equation:

d = ∫v(t)dt

= ∫−4t+6 m/sc dt

To calculate the integral, we need to find the antiderivative of −4t+6 m/sc, which is:

−4t−6m/sc+C

To find the constant C, we can use the fact that v(0) = 6 m/sec. So:

−4(0)−6m/sc+C=6 m/sc

So C = 12 m/sc. Therefore, the integral is:

∫−4t+6 m/sc dt = (1/2)(-4t)−(1/2)6m/sc+12m/sc = 2t−6m/sc

So the total distance traveled by the particle over the interval [0,5] is:

d = ∫v(t)dt = 2t−6m/sc |0→5 = 2(5)−6m/cs−2(0)−6m/sc+12m/sc = 60−30+12 = 42 m