# PhysicsGeneral motion in a straight line.

#### skeeter

##### Well-known member
MHB Math Helper
• Shah 72

#### Shah 72

##### Active member
$\displaystyle \dfrac{1}{12}\int_0^T 3t^3-16t^2-10t+60 \, dt = 0$

$T = \dfrac{\sqrt{1105}+5}{9} \approx 4.25$

in the attached graph of velocity vs time ...

shaded area above the x-axis = displacement from A to B

shaded area below the x-axis = displacement from B to A

View attachment 11188
Thanks so much!

#### Shah 72

##### Active member
$\displaystyle \dfrac{1}{12}\int_0^T 3t^3-16t^2-10t+60 \, dt = 0$

$T = \dfrac{\sqrt{1105}+5}{9} \approx 4.25$

in the attached graph of velocity vs time ...

shaded area above the x-axis = displacement from A to B

shaded area below the x-axis = displacement from B to A

View attachment 11188
$\displaystyle \dfrac{1}{12}\int_0^T 3t^3-16t^2-10t+60 \, dt = 0$

$T = \dfrac{\sqrt{1105}+5}{9} \approx 4.25$

in the attached graph of velocity vs time ...

shaded area above the x-axis = displacement from A to B

shaded area below the x-axis = displacement from B to A

View attachment 11188
Can you pls tell me how did you calculate that? Did you solve the cubic equation? I don't understand how you got the equation sq root (1105).

#### skeeter

##### Well-known member
MHB Math Helper
I integrated the velocity function and evaluated it from 0 to T and set the result equal to zero.

$\dfrac{T}{12}\left(\dfrac{3T^3}{4} - \dfrac{16T^2}{3} - 5T+ 60\right) = 0$

ignored the T/12 factor & multiplied the terms inside the parentheses by 12 to clear the fractions …

$9T^3 - 64T^2 -60T+720 = 0$

I graphed the cubic on my calculator and found T = 6 was a zero, then used synthetic division to find the quadratic factor …

$(T-6)(9T^2-10T-120) = 0$

$T = \dfrac{10 \pm \sqrt{4420}}{18}$

discarding the negative value for T …

$T = \dfrac{10+2\sqrt{1105}}{18} = \dfrac{5+\sqrt{1105}}{9}$

• Shah 72

#### Shah 72

##### Active member
I integrated the velocity function and evaluated it from 0 to T and set the result equal to zero.

$\dfrac{T}{12}\left(\dfrac{3T^3}{4} - \dfrac{16T^2}{3} - 5T+ 60\right) = 0$

ignored the T/12 factor & multiplied the terms inside the parentheses by 12 to clear the fractions …

$9T^3 - 64T^2 -60T+720 = 0$

I graphed the cubic on my calculator and found T = 6 was a zero, then used synthetic division to find the quadratic factor …

$(T-6)(9T^2-10T-120) = 0$

$T = \dfrac{10 \pm \sqrt{4420}}{18}$

discarding the negative value for T …

$T = \dfrac{10+2\sqrt{1105}}{18} = \dfrac{5+\sqrt{1105}}{9}$
Thank you so much!!