A car is moving with speed 30 m/sec on a circular path of radius 500 m. Its speed is increasing at the rate of 2m/sec². What is the acceleration of the car

The correct option is 2.7 m/s².

To find the total acceleration of the car moving along a circular path, we must account for two perpendicular components of acceleration: the tangential acceleration (which changes the speed) and the centripetal or radial acceleration (which changes the direction of motion).

1. Tangential Acceleration ($a_t$):
The rate at which the car's speed is increasing is the tangential acceleration. From the question, this is:
$a_t=2{\text{m/s}}^2$

2. Centripetal (Radial) Acceleration ($a_c$):
Centripetal acceleration is directed toward the center of the circular path and is calculated using the formula:
$a_c=\frac{v^2}{R}$
where:
- Speed of the car, $v=30\text{m/s}$
- Radius of the path, $R=500\text{m}$

Substituting these values into the formula:
$a_c=\frac{{30}^2}{500}=\frac{900}{500}=1.8{\text{m/s}}^2$

3. Total Acceleration ($a$):
Since the tangential acceleration vector is tangent to the circle and the centripetal acceleration vector points toward the center, they are perpendicular (at an angle of 90 degrees to each other). The magnitude of the net acceleration is the vector sum of these two components:
$a=\sqrt{{a_t}^2+{a_c}^2}$

Substituting the values:
$a=\sqrt{2^2+{1.8}^2}$
$a=\sqrt{4+3.24}$
$a=\sqrt{7.24}$
$a\approx 2.69{\text{m/s}}^2\approx 2.7{\text{m/s}}^2$

Therefore, the total acceleration of the car is 2.7 m/s².