Final answer:
The time (t) when velocity (v) is zero is when the projectile reaches its peak height. The velocities at t = 4 s and t = 5 s are negative, indicating downward movement after the peak. The motion and flight time of a projectile depend on the initial vertical velocity and the landing height.
Step-by-step explanation:
Projectile Motion Calculations
The given function for the distance s as a function of time t is s = 4[a]t - 4[b]t², with a=2 and b=7. This represents the vertical motion of a projectile, with the upward velocity and acceleration due to gravity taken into account.
Finding the Time (t) for Zero Velocity (v)
The velocity v at a time t is the derivative of the displacement with respect to time. We differentiate the given equation to find v.
v(t) = 8t - 8[b]t,
where [b] = 7.
Now, to find t for v = 0, we set the velocity equation to 0 and solve for t:
0 = 8t - 8[7]t
0 = 8t - 56t
0 = -48t
t = 0 s (at the point of launch)
Since a projectile has zero velocity at its maximum height as well, there is another value of t where v = 0, which is the time taken by the projectile to reach its maximum height.
Finding Velocity (v) for Specific Times
v(4) = 8(4) - 56(4) = 32 - 224 = -192 m/s
v(5) = 8(5) - 56(5) = 40 - 280 = -240 m/s
These negative velocities indicate that the projectile is moving downwards, as it would be after reaching its maximum height and starting to fall back down to earth.
Conclusions
This analysis of a projectile's motion demonstrates that velocity reaches zero at the peak of its trajectory and then becomes negative as it falls back down, representing the directional change towards the ground.
Additionally, the time of flight (or the time taken to reach the ground from the maximum point) for a projectile launched vertically is determined by its initial vertical velocity and the height at which it lands, as demonstrated through various examples provided.