A car is traveling along a road, and its engine is turning over with an angular velocity of +199 rad/s. The driver steps on the accelerator, and in a time of 14.2 s the angular velocity increases to +263 rad/s.
(a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of +199 rad/s during the entire 14.2-s interval?
(b) What would have been the angular displacement if the angular velocity had been equal to its final value of +263 rad/s during the entire 14.2-s interval?
(c) Determine the actual value of the angular displacement during the 14.2-s interval.
Define anguldi velocity and give lts units
(a) If the angular velocity had remained constant at +199 rad/s during the entire 14.2 s interval, we can calculate the angular displacement using the formula:
Angular displacement = angular velocity x time
Angular displacement = +199 rad/s x 14.2 s
But why did the car's engine only want to go to parties with right angles? It didn't want to be too acute or obtuse!
(b) If the angular velocity had been equal to its final value of +263 rad/s during the entire 14.2 s interval, we can use the same formula:
Angular displacement = angular velocity x time
Angular displacement = +263 rad/s x 14.2 s
But why was the car's engine always so cheerful? It had great rev-it-allity!
(c) To determine the actual value of the angular displacement during the 14.2 s interval, we need to consider the change in angular velocity. We can calculate it using the formula:
Change in angular velocity = final angular velocity - initial angular velocity
Change in angular velocity = +263 rad/s - +199 rad/s
Then, we can use this change in angular velocity, along with the time, to calculate the actual angular displacement using the formula:
Angular displacement = (initial angular velocity + final angular velocity) / 2 x time
Angular displacement = (+199 rad/s + 263 rad/s) / 2 x 14.2 s
But why was the car's engine always so philosophical? It would always ask, "What's my purpose? To drive you crazy!"
To solve these problems, we need to use the relationship between angular velocity (ω), time (t), and angular displacement (θ) given by the equation:
θ = ω * t
where θ is in radians, ω is in rad/s, and t is in seconds.
(a) To find the angular displacement if the angular velocity remained constant at the initial value of +199 rad/s during the entire 14.2-s interval, we can simply plug in the values into the equation:
θ = ω * t
θ = 199 rad/s * 14.2 s
θ = 2821.8 radians
So, the angular displacement would have been 2821.8 radians.
(b) To find the angular displacement if the angular velocity had been equal to its final value of +263 rad/s during the entire 14.2-s interval, we can use the same equation:
θ = ω * t
θ = 263 rad/s * 14.2 s
θ = 3734.6 radians
Therefore, the angular displacement would have been 3734.6 radians.
(c) Finally, to determine the actual value of the angular displacement during the 14.2-s interval, we need to consider the change in angular velocity and calculate the average angular velocity.
First, let's find the change in angular velocity:
Δω = final angular velocity - initial angular velocity
Δω = 263 rad/s - 199 rad/s
Δω = 64 rad/s
Now, let's calculate the average angular velocity:
Average ω = Δω / 2
Average ω = 64 rad/s / 2
Average ω = 32 rad/s
Using the equation θ = ω * t with the average angular velocity:
θ = average ω * t
θ = 32 rad/s * 14.2 s
θ = 454.4 radians
Therefore, the actual value of the angular displacement during the 14.2-s interval is 454.4 radians.
a. d = V*t = 199 rad/s * 14.2 s =
2825.8 rad.
b. d = V*t = 263 rad/s * 14.2 s = 3734.6 rad.
c. a = (263 - 199) / 14.2 = 4.51 rad/s^2.
d = at^2 = 4.51 * (14.2)^2 = 908.8 rad.