A stationary 1.67 kg object is struck by a stick. The object experiences a horizontal force given b F = at-bt^2, where t is the time in milliseconds from the instant the stick first contacts the object. If a = 1500 N/ms and b = 20N/ms^3, what is the speed of the object just after it comes away from the stick (t=2.74ms)?
Based upon the dimensions of b and the value of t when the object comes away, I believe you have copied the problem incorrectly. It should be
F = at-bt^3
in which case F = 0 at t = 2.74 ms
Integrate F dt from t=0 to t=2.74 ms
Express the answer in Newton*seconds
(not Newton*milliseconds)
That will equal the momentum of the released ball.
Use that and the mass to get the velocity.
3.29
To find the speed of the object just after it comes away from the stick, we need to integrate the force equation with respect to time to get the change in momentum. Then we can use this change in momentum to calculate the speed.
First, let's find the change in momentum. The equation for force F as a function of time t is given as F = at - bt^2, where a = 1500 N/ms and b = 20N/ms^3.
To integrate the force equation, we need to find the antiderivative of each term.
∫F dt = ∫(a - bt) dt
= ∫adt - ∫bt^2 dt
Integrating each term:
∫adt = at + C1 (where C1 is the constant of integration)
∫bt^2 dt = (1/3)bt^3 + C2 (where C2 is the constant of integration)
Combining the integrated terms:
∫F dt = at + C1 - (1/3)bt^3 - C2
Now, to find the change in momentum, we evaluate this expression at t = 2.74 ms and t = 0 (initial time).
Change in momentum = (∫F dt)2.74 - (∫F dt)0
= (a(2.74) + C1 - (1/3)b(2.74)^3 - C2) - (a(0) + C1 - (1/3)b(0)^3 - C2)
= a(2.74) - (1/3)b(2.74)^3
Now, let's substitute the given values of a and b:
Change in momentum = 1500(2.74) - (1/3)20(2.74)^3
= 4110 - (1/3)20(2.74)^3
= 4110 - (1/3)20(19.3568)
= 3761.224N·ms
The change in momentum is 3761.224 N·ms.
Finally, to find the speed of the object just after it comes away from the stick, we use the equation:
Change in momentum = mass * change in velocity
Since the object is stationary initially, the initial velocity is 0.
Change in momentum = mass * final velocity
Rearranging the equation:
final velocity = change in momentum / mass
Plugging in the values:
final velocity = 3761.224 N·ms / 1.67 kg
= 2252.092 m/s
Therefore, the speed of the object just after it comes away from the stick is approximately 2252.092 m/s.