A car starts from rest on a curve with a radius of 150 m and accelerates at 1.50 m/s^2. How many revolutions will the car have gone through when the magnitude of its total acceleration is 3.20 m/s^2 ?

total acceleration= sqrt(1.5^2+(v^2/r)^2)

3.2^2-1.5^2=(v^2/r)^2

solve for v^2/r

then,

angvelocity^2=2*angularacceleration*angulardisplacement

(V^2/r)1/r=2*1.50/r*angulardisplacement
solve for angular displacement, divide by 2PI to get revolutions

To solve this problem, we need to determine the speed of the car when the magnitude of its total acceleration is 3.20 m/s^2. We can do this by using the centripetal acceleration equation:

a_c = v^2 / r

Where:
a_c is the centripetal acceleration
v is the velocity of the car
r is the radius of the curve

Rearranging the equation, we can solve for v:

v^2 = a_c * r

Substituting the given values:

v^2 = 3.20 m/s^2 * 150 m

v^2 = 480 m^2/s^2

Taking the square root of both sides, we get:

v = √480 m/s

v ≈ 21.91 m/s

Now, we need to find the time it takes for the car to reach this speed. We can use the kinematic equation:

v = u + at

Where:
v is the final velocity
u is the initial velocity (0 m/s in this case since the car starts from rest)
a is the acceleration (1.50 m/s^2 in this case)
t is the time

Rearranging the equation, we can solve for t:

t = (v - u) / a

Substituting the given values:

t = (21.91 m/s - 0 m/s) / 1.50 m/s^2

t ≈ 14.61 s

Next, we need to find the number of revolutions the car has gone through in this time. We can use the formula:

θ = ω * t

Where:
θ is the angle in radians
ω is the angular velocity
t is the time

The angular velocity is directly related to the linear velocity by the equation:

ω = v / r

Substituting the given values:

ω = 21.91 m/s / 150 m

ω ≈ 0.146 rad/s

Now, we can calculate the angle by multiplying the angular velocity by time:

θ = 0.146 rad/s * 14.61 s

θ ≈ 2.134 radians

Finally, we need to convert the angle from radians to revolutions. Since there are 2π radians in one revolution, we can use this conversion factor:

1 revolution = 2π radians

θ in revolutions = θ in radians / 2π

θ in revolutions = 2.134 radians / (2π)

θ in revolutions ≈ 0.34 revolutions

Therefore, the car will have gone through approximately 0.34 revolutions when the magnitude of its total acceleration is 3.20 m/s^2.