Final answer:
The magnitude of a car's acceleration while traversing a circular path is calculated by combining centripetal and tangential accelerations using the Pythagorean theorem. Centripetal acceleration is determined by the car's velocity squared divided by the radius of the path, while tangential acceleration equals the rate of deceleration or acceleration along the path.
Step-by-step explanation:
The magnitude of acceleration of a car traveling along a circular path includes both the centripetal acceleration, which is due to the change in direction of the velocity vector, and the tangential acceleration, which is due to the change in the speed of the car. To find the total acceleration, one needs to consider both these components. The centripetal acceleration (ac) is given by the formula ac = v2/r, where v is the velocity and r is the radius of the circular path. The tangential acceleration (at) is simply the rate at which the speed of the car changes, and is provided in the problem statement.
To find the total acceleration (atotal), we use the Pythagorean theorem since the centripetal and tangential accelerations are perpendicular to each other:
atotal = √(ac2 + at2)
For example, if a car is decelerating at a rate of 9.0 km/h each second, this is equivalent to (9.0 km/h) / (3.6 km/h/s) = 2.5 m/s2 in SI units. If it is moving at 60.0 km/h (which is 16.67 m/s) along a circular path with a radius of 150.0 m, the centripetal acceleration would be (16.67 m/s)2 / 150.0 m = 1.85 m/s2. Thus, the magnitude of the total acceleration would be:
atotal = √(1.852 + 2.52) m/s2 = √(3.42 + 6.25) m/s2 = √9.67 m/s2 = 3.11 m/s2 (to three significant figures).