A shuffleboard disk is accelerated at a constant rate from...

A shuffleboard disk is accelerated at a constant rate from rest to a speed of $6.0 \mathrm{~m} / \mathrm{s}$ over a $1.8 \mathrm{~m}$ distance by a player using a cue. At this point the disk loses contact with the cue and slows at a constant rate of $2.5 \mathrm{~m} / \mathrm{s}^{2}$ until it stops. (a) How much time elapses from when the disk begins to accelerate until it stops? (b) What total distance does the disk travel?

Key Concepts

Constant Acceleration

This concept involves motion where an object’s velocity changes at a steady rate over time. It allows us to use specific kinematic equations to relate displacement, initial and final velocity, acceleration, and time, providing a straightforward framework for solving motion problems where the acceleration is uniform.

Deceleration (Negative Acceleration)

Deceleration refers to a reduction in an object’s speed, which is essentially acceleration in the direction opposite to the motion. In analysis, treating deceleration as negative acceleration lets us apply the same kinematic equations as for constant acceleration, with careful attention to the sign conventions for acceleration and displacement.

Kinematic Equations

These are a set of equations that describe the relationships between displacement, velocity, acceleration, and time in uniformly accelerated motion. They serve as the mathematical foundation for solving problems where an object experiences constant acceleration or deceleration, simplifying the calculation of unknown variables.

Piecewise Motion Analysis

When an object's movement is characterized by different phases, each with its own acceleration or deceleration, the overall motion is analyzed as separate segments. By breaking the motion into parts and applying the kinematic principles to each segment individually, we can calculate cumulative quantities like total time and overall distance traveled.