A car starts from rest on a curve with a radius of 190 and accelerates at 0.300 . How many revolutions will the car have gone through when the magnitude of its total acceleration is 2.00 ?
You should know by now that you have to include dimensions with your radius and acceleration numbers.
If the tangential acceleration is always 0.300 (m/s^2?), the centripetal accleration (which is perpendicular to tangential) will be 1.98 m/s^2 when the total acceleration is 2.00 m/s^2. From that, you can figure out the instantaneous velocity and how long it took to reach that speed.
To solve this problem, we can use the equations for centripetal acceleration and total acceleration, and then equate them to find the number of revolutions.
1. Centripetal acceleration (ac) is given by the equation:
ac = (v^2) / r
where v is the velocity and r is the radius of the curve.
2. Total acceleration (at) is the magnitude of the resultant acceleration vector and is given by:
at = √((ax^2) + (ay^2))
where ax and ay are the horizontal and vertical components of acceleration.
From the given information, we know:
Radius (r) = 190 m
Centripetal acceleration (ac) = 0.300 m/s²
Total acceleration (at) = 2.00 m/s²
To find the magnitude of the total acceleration vector, we can use the Pythagorean theorem:
at = √((ax^2) + (ay^2))
Since the car is starting from rest, the initial velocity (v) is 0. Therefore, the centripetal acceleration (ac) is equal to the total acceleration (at):
ac = at
Plugging in the values:
0.300 = (v^2) / 190
Rearranging the equation, we can solve for v:
(v^2) = 0.300 * 190
v^2 = 57
v ≈ 7.55 m/s
Now we can calculate the number of revolutions (N) using the formula:
N = (distance traveled) / (circumference of the curve)
The distance traveled is the circumference of the curve. The circumference (C) can be calculated as:
C = 2πr
Substituting the values:
C = 2π(190)
C ≈ 1193.6 m
The distance traveled is equal to the circumference of the circle. The distance traveled (d) is given by:
d = 2πrN
Substituting the values:
1193.6 = 2π(190)N
Now we can solve for N:
N ≈ 1193.6 / (2π(190))
N ≈ 1.00
Therefore, the car will have gone through approximately 1 revolution when the magnitude of its total acceleration is 2.00 m/s².