To determine the duration for which the force was applied, we can use the concept of impulse. Impulse is defined as the change in momentum of an object and is given by the product of the force and the time interval over which it acts.
The initial momentum of the object is given by the product of its mass and velocity:
Initial momentum (p_initial) = mass (m) * velocity (v) = 0.40 kg * 30 m/s = 12 kg·m/s
Since the force is applied until the velocity is reversed, the final momentum will be the negative of the initial momentum:
Final momentum (p_final) = -12 kg·m/s
To find the change in momentum, we calculate the difference between the final and initial momenta:
Change in momentum = p_final - p_initial = (-12 kg·m/s) - (12 kg·m/s) = -24 kg·m/s
The impulse (J) is equal to the change in momentum and is given by the product of the force and the time interval:
Impulse (J) = Force (F) * time (t)
In this case, the force is 2.0 N. Therefore:
-24 kg·m/s = 2.0 N * t
Simplifying the equation, we can solve for the time (t):
t = (-24 kg·m/s) / (2.0 N) = -12 s
The negative sign indicates that the force was applied in the opposite direction of the object's initial velocity. However, time cannot be negative in this context. Therefore, we discard the negative sign, resulting in:
t = 12 s
Hence, the force was applied for 12 seconds.