At t = 0, an automobile traveling north begins to make a turn. It follows one-quarter of the arc of a circle of radius 10.4 m until, at t = 1.66 s, it is traveling east. The car does not alter its speed during the turn.
To solve this problem, we can break it down into several steps.
Step 1: Determine the initial and final velocities of the car.
Given that the car is initially traveling north and ends up traveling east, we can infer that the initial velocity is purely in the north direction and the final velocity is purely in the east direction.
Step 2: Calculate the change in velocity (∆v) and change in time (∆t).
Since the car does not alter its speed during the turn, the magnitude of the change in velocity (∆v) is equal to the magnitude of the initial velocity. The change in time (∆t) is given as 1.66 s.
Step 3: Apply the equations of motion to solve for the angle of the arc (∆θ).
Using the equation v = ∆θ/∆t, where v is the magnitude of the velocity, we can rearrange the equation to get ∆θ = v * ∆t. However, since we know that the car traveled one-quarter of the arc of a circle, we need to divide ∆θ by 4 to get the angle for this specific arc.
Step 4: Calculate the radius of the circle.
Using the relationship between the angle of the arc (∆θ) and the radius (r), we have the equation ∆θ = s/r, where s is the arc length. Since we know that the traveled arc is one-quarter of the circle, we can write ∆θ as π/2 radians. Solving for r gives us r = s / (∆θ).
Step 5: Calculate the radius and substitute it into the equation obtained in step 4 to find s, the arc length.
Substituting the known values into the equation r = s / (∆θ), we can solve for s.
Step 6: Calculate the values for the radius and the arc length.
Substitute the values obtained in steps 4 and 5 into the respective equations to find the radius and the arc length.
By following these steps, you can determine the radius of the circle and the length of the arc that the car travels.