Final answer:
To find the distance and time before a particle stops when starting with an initial velocity and subjected to a velocity-dependent deceleration, one must integrate the velocity over time using the relationship dv/dt = a and accounting for initial conditions.
Step-by-step explanation:
To determine how far a particle travels before it stops and the time it takes, given it has an initial velocity and a velocity-dependent deceleration, we can use kinematic equations and calculus. The given deceleration is a = -1.4 √ v, and the initial velocity is 9 m/s.
To find the distance traveled before the particle stops, we use the relation dv/dt = a, where v is velocity and t is time. Substituting the given deceleration:
dv/dt = -1.4 √ v
We separate variables and integrate:
∫ dv / √ v = -1.4 ∫ dt
2 √ v = -1.4t + C
Using the initial velocity to find C, we determine the time when v=0 m/s.
For distance, we then integrate velocity over time:
∫ v dt = ∫ 0.49(2 √ v) dt
We can find an expression for time as a function of velocity from our previous calculation and substitute in:
∫ 0.49(2 √ v) (dt/dv) dv
Calculating this integral gives us the total distance traveled until the particle stops. We now have both the time it takes for the particle to stop and the distance it travels.