Question

A particle moves along the x-axis so that at time

t (in seconds), the position of the particle is defined by x(t)=(2/3)(t^3)-3t^2+4t centimeters.

a) during 0≤t≤4 seconds, what time interval(s) is the particle moving right? Justify your answer.

b) what's the particle's maximum velocity on the interval 0≤t≤4 seconds and what time does it occur? Justify your answer.

c) What's the particle's maximum acceleration on the interval 0≤t≤4 seconds and what time does it occur? justify your answer.

d) During 0≤t≤4 seconds, what time interval(s) is the particle speeding up? Justify your answer.

PLEASE show your steps on paper and if possible give explanations on how you came to solve each question.

Answer 1

Answer: a) To determine when the particle is moving right, we need to find when its velocity is positive. The velocity of the particle is given by the derivative of its position function:

v(t) = 2t^2 - 6t + 4

We can factor this quadratic expression to get:

v(t) = 2(t - 1)(t - 2)

So the velocity is positive for t > 2 and t < 1. Therefore, the particle is moving right during the time interval 1 < t < 2.

b) The velocity function v(t) found in part (a) is a quadratic function with a negative leading coefficient, which means that it is maximized at its vertex. The vertex of the quadratic function is given by:

t = -b / (2a)

where a = 2, b = -6. Plugging in these values, we get:

t = -(-6) / (2 * 2) = 1.5

Therefore, the maximum velocity occurs at t = 1.5 seconds. To find the maximum velocity, we can plug this value of t into the velocity function:

v(1.5) = 2(1.5)^2 - 6(1.5) + 4 = -1

So the maximum velocity is -1 cm/s.

c) The acceleration of the particle is given by the derivative of its velocity function:

a(t) = 4t - 6

This is a linear function with a positive slope, which means that the acceleration is constant and increasing. The maximum acceleration occurs at t = 4 seconds, and its value is:

a(4) = 4(4) - 6 = 10

So the maximum acceleration is 10 cm/s^2.

d) The particle is speeding up when its acceleration is positive. From part (c), we know that the acceleration is positive for all t in the interval 0 ≤ t ≤ 4, so the particle is speeding up during this entire time interval.

Explanation: