A 109 kg fullback runs at the line of scrimmage.

(a) Find the constant force that must be exerted on him to bring him to rest in a distance of 1.1 m in a time interval of 0.22 s.
(b) How fast was he running initially?

Average V during deceleration = 1.1/0.22 = 5 m/s

Initial V = Vo = 2*Vav = 10 m/s
Deceleration rate = a = Vo/0.22s
= 45.5 m/s^2
Force = m*a

(a) To find the constant force required to bring the fullback to rest in a distance of 1.1 m in a time interval of 0.22 s, we can use the equation:

F = (m * Δv) / Δt.

Where:
F is the force,
m is the mass of the fullback (109 kg),
Δv is the change in velocity,
Δt is the time interval.

Since the fullback comes to rest, the change in velocity is equal to the initial velocity. Therefore, Δv equals the initial velocity.

Using the equation, we can rearrange it to solve for the force:

F = (m * Δv) / Δt
F = (109 kg * Δv) / 0.22 s
F = 496 kg * Δv / s.

Now, we need to find the value of Δv. We can use the equation for linear motion:

Δv = (vf - vi).

Since the fullback comes to rest, the final velocity (vf) is 0. Therefore:

Δv = (0 - vi) = -vi.

Substituting this into the force equation:

F = 496 kg * (-vi) / s
F = -496 vi kg / s.

Therefore, the constant force required to bring the fullback to rest is -496 vi kg / s, where vi is the initial velocity.

(b) To find the initial velocity, we can rearrange the force equation:

F = -496 vi kg / s
vi = -F * s / (496 kg).

Substituting the given values: F = -496 vi kg / s, Δs = 1.1 m, and Δt = 0.22 s:

vi = -F * s / (496 kg)
vi = -(-496 vi kg / s) * 1.1 m / (496 kg)
vi = (-496 vi m / s^2) * 1.1 m
vi = 544.6 vi m / s^2

Simplifying:

vi = 544.6 vi.

To find the initial velocity, we divide both sides by vi:

vi / vi = 544.6.

Therefore, the initial velocity of the fullback is 544.6 m/s.

To find the force exerted on the fullback, we can use the equation:

Force (F) = (mass (m) * change in velocity (Δv)) / time interval (Δt)

(a) To find the force, we need to calculate the change in velocity first. The final velocity (vf) is 0 m/s since the fullback comes to rest, and the initial velocity (vi) is what we need to find.

Using the equation:

vf = vi + (acceleration * time)

Since the acceleration is constant and the fullback comes to rest, vf = 0 m/s:

0 m/s = vi + (acceleration * time)

Rearranging the equation:

vi = - (acceleration * time)

Now, we can substitute the values into the equation:

vi = - (acceleration * time)
vi = - [(Δv) / Δt] * Δt
vi = - Δv

Thus, the initial velocity (vi) is equal to the change in velocity (Δv) with a negative sign.

Now, we know that the change in velocity (Δv) is given by:

Δv = vf - vi

In this case, Δv = 0 - vi = -vi.

Now, substitute the values into the force equation:

Force (F) = (mass (m) * change in velocity (Δv)) / time interval (Δt)
F = (mass (m) * (-vi)) / Δt
F = (109 kg * (-vi)) / 0.22 s
F = - 109 kg * (vi / 0.22 s)

The constant force exerted on the fullback to bring him to rest in a distance of 1.1 m in a time interval of 0.22 s would be - 109 kg * (vi / 0.22 s).

(b) To find the initial velocity (vi), we can rearrange the equation:

vi = - (acceleration * time)
vi = - [(Δv) / Δt] * Δt
vi = - Δv

Substituting the values:

vi = -Δv
vi = - (0 m/s)

Therefore, the initial velocity (vi) is 0 m/s.