91Ó°ÊÓ

When an object moves in a circular path, even if it maintains constant speed, its direction is constantly changing. This change in direction is due to radial or centripetal acceleration. In simple terms, radial acceleration acts towards the center of the circular path.
This kind of acceleration is responsible for keeping the object on its curved path. To find radial acceleration, we use the formula:

• \( a_r = \frac{v^2}{r} \)

Here, \( v \) represents the object's velocity, and \( r \) is the radius of the circle. For example, if the velocity is \( 4.00 \, \text{m/s} \) and the circle's radius is \( 10.0 \, \text{m} \), the radial acceleration will be \( 1.60 \, \text{m/s}^2 \). Remember, radial acceleration is always directed inward, towards the circle’s center.

While radial acceleration focuses on changing the direction of movement, tangential acceleration looks at changing the speed along the tangent of the path. It's a linear acceleration and occurs when the speed of the object increases or decreases.
To find tangential acceleration, we use the derivative of the velocity with respect to time:

• \( a_t = \frac{dv}{dt} \)

For the stuntman, whose velocity is given by \( v = c t^2 \), taking the derivative results in:

• \( a_t = 2c t \)

Substituting \( c = 1.00 \, \text{m/s}^3 \) and \( t = 2.00 \, \text{s} \), the tangential acceleration becomes \( 4.00 \, \text{m/s}^2 \).Unlike radial acceleration, tangential acceleration can point forward or backward, depending on whether the object's speed is increasing or decreasing along the path.

Total acceleration is the combination of both radial and tangential accelerations. It is a vector quantity, meaning it has both magnitude and direction.
To calculate the magnitude of total acceleration, use the Pythagorean theorem as both components are perpendicular to each other:

• \( a_{\text{total}} = \sqrt{a_r^2 + a_t^2} \)

Given \( a_r = 1.60 \, \text{m/s}^2 \) and \( a_t = 4.00 \, \text{m/s}^2 \), this results in:

• \( a_{\text{total}} = 4.31 \, \text{m/s}^2 \)

To find the angle between the total acceleration and radial acceleration, we use:

• \( \tan \theta = \frac{a_t}{a_r} \)

This equation gives the angle \( \theta = 68.2^\circ \) between the acceleration vectors. This angle tells us how the total acceleration vector is oriented relative to the radial direction, providing a clearer picture of motion dynamics.