The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by the expression v = (-5.15 multiplied by 107) t2 + (2.30 multiplied by 105) t, where v is in meters per second and t is in seconds. The acceleration of the bullet just as it leaves the barrel is zero.

BDetermine the length of time the bullet is accelerated.

CFind the speed at which the bullet leaves the barrel.

DWhat is the length of the barrel?

From the definition of acceleration we know thata=dvdtTaking the derivative of v with respect to t yieldsa=dvdt=(−5.15∗107)∗2∗t+(2.30∗105)
From the definition of distance, we know that v=dxdt→dx=vdtIntegrating velocity yieldsx=(−5.15∗107)∗(t33)+(2.30∗105)∗(t22)+x0 where x0 is the starting position.

If acceleration is zero when the bullet leaves the barrel, we can use our equation for acceleration to determine the time the bullet is in the barrel. This is seen as0=(−5.15∗107)∗2∗t+(2.30∗105)→t=−(2.30∗105)(−5.15∗107)

Knowing time, we can solve for the velocity as the bullet leaves the barrel by plugging time back into our given equation for velocity.

The length of the barrel can be solved by plugging time back into our equation for distance and setting x0=0.