What is the time required for a freight train's wheels, currently at 38 rad/s, to accelerate to a speed 1.5 times faster with an angular acceleration of 0.80 rad/s²?

To find the time required for the freight train's wheels to accelerate from 38 rad/s to a speed that is 1.5 times faster, we first need to calculate the final angular speed. This is done by multiplying the initial speed by 1.5:

Final angular speed = 1.5 × 38 rad/s = 57 rad/s.

Next, we can use the formula for angular acceleration to find the time it takes to reach this final angular speed. The formula is:

[

\alpha = \frac{\Delta \omega}{\Delta t}

]

Where:

• (\alpha) is the angular acceleration (0.80 rad/s²),

• (\Delta \omega) is the change in angular speed (final speed - initial speed), and

• (\Delta t) is the time taken (which we're solving for).

First, calculate the change in angular speed:

(\Delta \omega = \text{final speed} - \text{initial speed} = 57 rad/s - 38 rad/s = 19 rad/s).

Now, substitute the known values into the angular acceleration formula:

[

0.80 \text{ rad/s}^2 = \