a. For the same speed, the acceleration of the object varies _____________ (directly, inversely) with the radius of curvature.

Answer

The acceleration of the object varies inversely with the radius of curvature for the same speed.

This relationship can be understood through the concept of centripetal acceleration, which is defined as:

$a_c = \frac{v^2}{r}$

where:

• $a_c$ is the centripetal acceleration,
• $v$ is the speed (velocity) of the object,
• $r$ is the radius of curvature.

From this formula, it's clear that as the radius $r$ increases, the centripetal acceleration $a_c$ decreases if the speed $v$ remains constant, illustrating that acceleration varies inversely with the radius of curvature. Thus, if one were to increase the radius of curvature, the acceleration required to maintain the same speed would decrease, confirming the inverse relationship.