A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of a = ( -6t) m/s^2, where t is in seconds, determine the distance traveled before it stops.

Question

A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of a = ( -6t) m/{ s }^{ 2 }, where t is in seconds, determine the distance traveled before it stops.

Accepted Answer

Velocity: [latex]{ v }_{ 0 } = 27 m/s ext{ at } { t }_{ 0 } = 0 s[/latex]. We have

[latex]\begin{aligned} (+ \downarrow) \quad\quad & dv = adt \ & \int_{ 27 }^{ v } dv = \int_{ 0 }^{ t } -6tdt \ & v = (27 - 3{ t }^{ 2 }) m/s \quad\quad\quad\quad (1) \end{aligned}[/latex]

At v = 0 , from Eq. (1)

[latex]0 = 27 - 3{ t }^{ 2 }\quad\quad t = 3.00s[/latex]

Distance Traveled: [latex]{ s }_{ 0 } = 0[/latex] m at [latex]{ t }_{ 0 } = 0s [/latex]. Using the result [latex]v = 27 - 3{ t }^{ 2 }[/latex], we have

[latex]\begin{aligned} (+ \downarrow) \quad\quad & ds = vdt \ & \int_{ 0 }^{ s } ds = \int_{ 0 }^{ t } (27 - 3{ t }^{ 2 })dt \ & s = (27t - { t }^{ 3 }) m \quad\quad\quad\quad (2) \end{aligned}[/latex]

At t = 3.00 s , from Eq. (2)

[latex]s = 27(3.00) - { 3.00 }^{ 3 } = 54.0m[/latex]

Question : A sphere is fired downwards into a medium with an initial sp...

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