Final answer:
To determine the time at which the cart reaches an acceleration of 10 m/s², we can use the equation for total acceleration and the given equation for the angular velocity. By substituting the value of t into the equation, we can solve for the angular velocity. We can then use this angular velocity to solve for t when the acceleration reaches 10 m/s².
Step-by-step explanation:
Given the equation b = -1, we can determine the angular velocity, in radians per second, at any time t by substituting the value of t into the equation.
To find the time at which the cart reaches an acceleration of 10 m/s², we need to determine the angular velocity at that time and then solve for t.
Since the total acceleration (a_total) is the sum of the tangential acceleration (a_t) and the centripetal acceleration (a_c), we can write:
a_total = a_t + a_c
Since the centripetal acceleration is given by the equation a_c = r * (angular velocity)^2, we can substitute the value of a_c and rearrange the equation to find the angular velocity at the time when a_total reaches 10 m/s².
Once we have the angular velocity, we can substitute it into the equation b = -1 and solve for t.
Therefore, the cart reaches an acceleration of 10 m/s² at the time t when b = -1.