For the pinball game of Prob. 2 / 109, if the plunger...

For the pinball game of Prob. $2 / 109$, if the plunger imparts an initial speed of $3 \mathrm{m} / \mathrm{s}$ to the ball at time $t=0,$ determine the acceleration a of the ball ( $a$ ) at time $t=0.08 \mathrm{s}$ and $(b)$ at time $t=0.20 \mathrm{s}$. At point $F$ the speed of the pinball has decreased by $10 \%$ from the initial value, and this decrease may be assumed to occur uniformly over the total distance traveled by the pinball. Use the values $r=150 \mathrm{mm}$ and $\theta=60^{\circ}$

Key Concepts

Centripetal (Radial) Acceleration

In circular or curved motion the velocity changes direction continuously even if its magnitude remains constant. Centripetal acceleration is the inward directed acceleration required to keep an object following a curved path, calculated by the formula a_c = v²/r. It depends on the instantaneous speed and the radius of the curvature of the path, and always acts perpendicular to the tangent of the motion.

Uniform Deceleration Over a Distance

This concept refers to a situation where the reduction in speed is spread out evenly with respect to the distance traveled. Instead of assuming a constant deceleration with respect to time, the decrease in speed is taken as uniform per unit distance. This allows one to relate the change in speed directly to the distance along the path via a constant rate (dv/ds), and then use the chain rule to connect this spatial rate of change to the time rate of change (acceleration).

Tangential Acceleration in Curved Motion

Tangential acceleration is responsible for changing the magnitude of the velocity of an object moving along a curved path. In problems where the speed decreases uniformly over the traveled distance, the tangential acceleration can be determined by multiplying the constant speed?decrease per distance by the instantaneous speed (using the chain rule: a_t = v·(dv/ds)). This component of acceleration acts along the tangent to the path and is distinct from any change in direction.

Non?uniform Circular Motion

Non?uniform circular motion occurs when an object moving on a circular path experiences a change in speed as well as a change in direction. In such cases, the total acceleration is the vector sum of the tangential component (which changes the speed) and the centripetal component (which changes the direction). Understanding this decomposition is essential to analyzing problems where speed reductions are specified along a circular arc, as well as determining the magnitude and direction of the overall acceleration at any given point.

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