A car starts from rest on a curve with a radius of 130m and accelerates at 1.1m/s^2 .Through what angle will the car have traveled when the magnitude of its total acceleration is 2.7m/s^2 ?

To find the angle through which the car will have traveled, we need to use the equation relating centripetal acceleration with radius and angular speed:

a_c = r * ω^2

where:
a_c = centripetal acceleration
r = radius of the curve
ω = angular speed (in radians per second)

First, we need to find the angular speed. To do that, we can use the equation that relates linear acceleration (a) with angular speed (ω) and radius (r):

a = r * α

where:
a = linear acceleration
r = radius of the curve
α = angular acceleration

From the given information, we have the linear acceleration (a = 2.7 m/s^2) and radius (r = 130 m). We also know that the car starts from rest, so the initial angular speed (ω_0) is 0.

Using the equation above, we can rearrange it to solve for angular acceleration (α):

α = a / r

Now, we can find the angular speed (ω) using the equation for angular motion:

ω = ω_0 + α * t

Since the car starts from rest, ω_0 = 0, and we can rearrange the equation to solve for time (t):

ω = α * t
t = ω / α

Now, we can substitute the values for α (found earlier) and ω_0, and solve for time (t).

Next, we can use our calculated time (t) and the initial angular speed (ω_0 = 0) to find the final angular speed (ω) using the equation:

ω = ω_0 + α * t

With the final angular speed (ω) found, we can use it to calculate the angle (θ) traveled by the car using the equation:

θ = ω * t

Now, let's calculate the answer step by step.

Step 1: Calculate the angular acceleration (α)
α = a / r
α = 2.7 m/s^2 / 130 m

Step 2: Calculate the time (t)
t = ω / α
t = ω / (2.7 m/s^2 / 130 m)

Step 3: Calculate the final angular speed (ω)
ω = α * t
ω = (2.7 m/s^2 / 130 m) * (ω / (2.7 m/s^2 / 130 m))

Step 4: Calculate the angle traveled (θ)
θ = ω * t
θ = (ω / (2.7 m/s^2 / 130 m)) * (ω / (2.7 m/s^2 / 130 m))