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.

Transcript

00:01 So far this over here that initially they are at rest okay so in ub equals to 0 and it is giving up to the velocity b equals to that is given as 6 meter per second for distance as equal to 1 .8 meter okay so first of all we will find the acceleration using given data that we know that v square equal to what u a square plus 2 a s right so v squared so v square equals to 6 square equal to u s so 2 as a will be 6 square that is 36 2 into s right so b squared so b s 2 x squared that is 36 2 into s 1 .8 so it will be 3 .6 so a will be 10 meter of per second square okay so now we can find time from here okay so how much time it haven't taken so v equal to u plus 80 we are going if this formula so u is 0 so t is nothing but 6 equal to 10 into t t equal to what 6 .10 0 .6 second okay now we need to find a total time okay so this is the 6 0 .6 second to speeding up to that speed now it is de - accelerating with the acceleration of equal to minus of 2 .5 meter second square because the acceleration means obviously this is minus sign and and it has gained the velocity of 6 meter per second squared so now we are going to use v equal to u plus a t and in this case final velocity is going to be 0 so t equal to minus of u upon a so u means six that is the initial velocity and minus of minus 2 .5 so it will be i guess 60 upon 25 so 12 upon 5 so 2 .4 okay 2 .4 second so how much time it have taken from a speeding up to slowing down so t equal to t1 plus 3 2 that means 0 .6 second plus 2 .4 so that means 3 seconds right in the next scenario what we need to find is we need to find that how much time elaps from the from when the disk begin to accelerate until stop okay and what is that distance travel okay so in the first half this have travel 1 .8 meter the second have you are going to use v .s square equal to u s right we have given final velocity is zero in its ability to u s equal to minus of u s square upon 2a so u means 36 square 36 upon minus of because acceleration is negative 2 .5 into 2 so that means 5 so minus and minus plus as equal to 36 .5 that means 7 .2 okay and the total distance d will be 7 .2 plus 1 .8 that will be nothing but 9 .8 1m...

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