The force on an object is described as a function of time equation:

F(t) = 3t^2+7t
What is the impulse delievered by the force over a period of time from 0 to 5.0s? What is the average force exerted on an object over this time period? If the object has a mass of 2.0kg and is initally at rest, what is the velocity of the object at 5.0s?

To find the impulse delivered by the force over a period of time from 0 to 5.0s, we need to calculate the definite integral of the force function F(t) over that time interval.

The formula for impulse is given by:
Impulse = ∫[t1, t2] F(t) dt

Since we are given the force function, F(t) = 3t^2 + 7t, we can substitute it into the integral expression:

Impulse = ∫[0, 5] (3t^2 + 7t) dt

To evaluate this integral, we need to integrate each term separately. Recall that the power rule for integration states that for any term of the form Ax^n, the integral is [(A/(n+1)) * x^(n+1)].

So, let's integrate 3t^2 and 7t separately:
∫(3t^2) dt = (3/3) * t^3 = t^3
∫(7t) dt = (7/2) * t^2 = (7/2) * t^2

Now we can evaluate the definite integral as follows:

Impulse = [t^3] from 0 to 5 + [(7/2) * t^2] from 0 to 5
= (5^3) - (0^3) + (7/2) * (5^2) - (7/2) * (0^2)
= 125 + (7/2) * 25
= 125 + 87.5
= 212.5 Ns

So, the impulse delivered by the force over the time period from 0 to 5.0s is 212.5 Ns (Newton-seconds).

To find the average force exerted on the object over this time period, we can use the formula:

Average Force = Impulse / Time

In this case, Impulse is 212.5 Ns and the time interval is 5.0s:

Average Force = 212.5 Ns / 5.0s
= 42.5 N

Therefore, the average force exerted on the object over the time period from 0 to 5.0s is 42.5 N (Newtons).

To determine the velocity of the object at 5.0s, we need to use the concept of impulse-momentum. The impulse delivered to an object is equal to the change in momentum of the object. The formula for impulse-momentum is:

Impulse = Change in Momentum

The momentum of an object is given by the product of its mass and velocity:

Momentum = Mass * Velocity

Given that the mass of the object is 2.0kg and the impulse is 212.5 Ns (from the previous calculation), we can rearrange the formula to solve for velocity:

Impulse = Mass * Velocity
212.5 Ns = (2.0kg) * Velocity

Now we can solve for the velocity:

Velocity = Impulse / Mass
Velocity = 212.5 Ns / 2.0kg
Velocity = 106.25 m/s

Therefore, the velocity of the object at 5.0s is 106.25 m/s (meters per second).