Question

a cart is moving along the ( x )-axis with its position, in meters, described by ( x = t^2 - 6t + 8 ), where ( t ) is the time, in seconds, in the interval ( 0 leq t leq 6 ) s. which of the following statements best describes the momentum of the cart in this time interval?

a the cart’s momentum changes direction at exactly one instant.

b the cart’s momentum changes direction at exactly two different instants.

c the cart’s momentum is constant.

d the cart’s momentum is not constant and does not change directions at any point in this interval.

Explanation:

Step1: Find velocity function

Velocity $v(t)$ is derivative of position $x(t)$.
$v(t) = \frac{dx}{dt} = \frac{d}{dt}(t^2 - 6t + 8) = 2t - 6$

Step2: Find velocity sign change

Set $v(t)=0$ to find direction change:
$2t - 6 = 0 \implies t=3$
Check sign: For $0\leq t<3$, $v(t)<0$; for $30$.

Step3: Relate to momentum

Momentum $p=mv$, so direction matches velocity.

Answer:

A. The cart's momentum changes direction at exactly one instant.