Solving an Accelerating Car's Revolutions

[jjones1573]

Homework Statement

A car is coming to a top over a distance of 96m

Vi = 20m/s
r = 25cm

find the intial angular velocity, angular acceleration, total revolutions

Homework Equations

w = v / r
angular acceleration = change in w / t
d = (Vi + Vf / 2) t

The Attempt at a Solution

So for the initial angular velocity I did w = v / r
= 20m/s / 0.25m
= 80 rad/s

then I calculated the time based on displacement d = (Vi + Vf / 2) t
therefore t = d / (Vi + Vf / 2)
= 96 / 10
= 9.6s

So that I could find angular acceleration = change in w / t
= 80 / 9.6
= 8.3 rad/s

I THINK that's right so far

and for revolutions I did 80rad/s / 2phi
to get 12.7 rev/s

but then this is only the rate of rev/s at t=0 . This would decrease as the car deccelerates right? So how can I find the overal number of revolutions?

 

[gneill]

So how can I find the overal number of revolutions?

If you assume that the wheel doesn't slip on the road surface, then you should be able to count the number of wheel circumferences that go into the total distance.

 

[jjones1573]

Ah of course! that makes sense. What about in an instance where a ball is moving through the air, how could you measure the revolutions?

 

[gneill]

Ah of course! that makes sense. What about in an instance where a ball is moving through the air, how could you measure the revolutions?

Without non-slip contact with a surface there's no obvious way to fix a relationship between distance and revolutions -- the wheel could have any rotational velocity independent of linear speed.

 

[rwisz]

I would like to point out that there are a few things to consider in this problem. First, the given information is insufficient to accurately solve for the initial angular velocity and angular acceleration. We do not have enough information about the car's mass, force, or any external factors that may affect its motion. Additionally, the given units are not consistent (m/s and cm), which could lead to errors in calculations. It is important to always use consistent units in scientific calculations.

However, assuming that the given information is accurate and there are no external factors affecting the car's motion, I will provide a general solution.

To find the initial angular velocity, we can use the equation w = v / r, as you have correctly done. However, we need to convert the radius from cm to m, so it would be 0.25m instead of 25cm. This gives us an initial angular velocity of 80 rad/s.

For the angular acceleration, we can use the equation a = (Vf - Vi) / t. Since the car is coming to a stop, Vf would be 0 m/s. Using the time calculated, we get an angular acceleration of -2.08 rad/s^2 (negative because the car is decelerating).

To find the total revolutions, we need to consider the initial and final angular velocities. The car starts at 80 rad/s and stops at 0 rad/s, so the average angular velocity would be 40 rad/s. We also know that the car travels a distance of 96m in 9.6 seconds. To find the total number of revolutions, we can use the formula d = wavg * t, where d is the distance traveled and wavg is the average angular velocity. This gives us 1536 revolutions.

However, as mentioned earlier, this solution is based on certain assumptions and may not accurately reflect the real-world situation. Further analysis and additional information would be needed to provide a more accurate and precise solution.