(a) Well, to find the initial angular speed of the tires, we can use the formula:
Angular speed = (final angular displacement) / (time)
Since the truck comes to rest, the final angular displacement is 49.0 turns, which is equal to 49.0 * 2Ï€ rad.
Now, we need to find the time it takes for the truck to come to rest. The truck slows uniformly, so we can use the equation:
Final angular speed = Initial angular speed + (angular acceleration) * (time)
Since the final angular speed is 0 (truck comes to rest), we can rearrange the equation to solve for time:
time = (Final angular speed - Initial angular speed) / (angular acceleration)
Now, we have all the information to find the initial angular speed:
Initial angular speed = (final angular displacement) / (time)
= (49.0 * 2Ï€ rad) / ((Final angular speed - Initial angular speed) / (angular acceleration))
But since we know the final angular speed is 0, we can simplify the equation:
Initial angular speed = (49.0 * 2Ï€ rad) / (time / (angular acceleration))
Well, this equation is going in circles! So, we need to rearrange it again and isolate the initial angular speed:
Initial angular speed = (49.0 * 2Ï€ rad) * (angular acceleration) / time
Now we can plug in the given values and solve for the initial angular speed.
(b) To find the angular acceleration of the tires, we can substitute the values into the equation:
Final angular speed = Initial angular speed + (angular acceleration) * (time)
Since the final angular speed is 0, we are left with:
0 = Initial angular speed + (angular acceleration) * (time)
Now, we can rearrange the equation to solve for angular acceleration:
angular acceleration = -Initial angular speed / time
Again, we can plug in the given values and solve for angular acceleration.
(c) To find the distance the truck travels before coming to rest, we can use the formula:
Distance = ((Initial angular speed) * (time) + (1/2) * (angular acceleration) * (time)^2) * (radius)
Since the truck comes to rest, the initial angular speed is 0.
So, the equation simplifies to:
Distance = (1/2) * (angular acceleration) * (time)^2) * (radius)
Now, plug in the given values and solve for distance.
But hey, while this calculation might not be a walk in the park, at least you'll have a tire-iffic time with all those turns!