[table]
In a robotics tournament, a contestant programs a battlebot’s spinning blade to speed up with a constant angular acceleration of 2.25 rad/s². After 10.0 seconds, the blade reaches an angular velocity of 35.0 rad/s. What was the blade’s initial angular velocity before it began to speed up?
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Question

BlackTom AI · Accepted Answer

We’re given a constant angular acceleration α = 2.25 rad/s², a time interval t = 10.0 s, and the final angular velocity ω = 35.0 rad/s. We can relate initial and final angular velocity with ω = ω0 + α t, where ω0 is the initial angular velocity before speeding up.

Option a: 3.75 rad/s. If ω0 were 3.75 rad/s, then the change in angular velocity over 10 s would be Δω = α t = 2.25 × 10 = 22.5 rad/s. That would give a final ω = ω0 + Δω = 3.75 + 22.5 = 26.25 rad/s, which does not match the given 35.0 rad/s. Therefore this option would imply the final speed is too small.

Option b: 12.5 rad/s. If ω0 were 12.5 rad/s, then ω = ω0 + α t = 12.5 + (2.25 × 10) = 12.5 + 22.5 = 35.0 rad/s, which matches the provided final angular velocity exactly. This is the correct consistency check for the given data.

Option c: 10.7 rad/s. Using ω0 = 10.7 rad/s leads to ω = 10.7 + 22.5 = 33.2 rad/s, which is not equal to the stated 35.0 rad/s. Hence this value would produce an incorrect final speed.

Option d: 0 rad/s. With ω0 = 0, ω = 0 + 22.5 = 22.5 rad/s, which is far from 35.0 rad/s. This clearly cannot be correct given the acceleration and time provided.

Option e: 6.25 rad/s. If ω0 = 6.25 rad/s, then ω = 6.25 + 22.5 = 28.75 rad/s, again not matching the given final speed of 35.0 rad/s. This option fails the data consistency test as well.

In summary, the only value of ω0 that makes the final angular velocity after 10.0 s with a constant acceleration of 2.25 rad/s² equal to 35.0 rad/s is 12.5 rad/s, since ω = ω0 + α t must hold true. The other options lead to final speeds that do not coincide with the provided final velocity when using the same acceleration and time.

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