To determine how long the divers take to reach the water's surface, we need to solve the following equation of motion for time [tex]\( t \)[/tex]:
[tex]\[ f(t) = \frac{1}{2} a t^2 + v_0 t + h \][/tex]
where:
- [tex]\( a = 9.8 \, \text{m/s}^2 \)[/tex] (acceleration due to gravity),
- [tex]\( v_0 = 3 \, \text{m/s} \)[/tex] (initial velocity),
- [tex]\( h = 20 \, \text{m} \)[/tex] (initial height of the platform).
We want to find the time [tex]\( t \)[/tex] when the divers reach the water’s surface, which means the height [tex]\( f(t) \)[/tex] will be 0 (the water level). Thus, we need to solve:
[tex]\[ 0 = \frac{1}{2} \cdot 9.8 \cdot t^2 + 3 \cdot t + 20 \][/tex]
This simplifies to:
[tex]\[ 0 = 4.9 t^2 + 3 t + 20 \][/tex]
This is a quadratic equation of the form [tex]\( a t^2 + b t + c = 0 \)[/tex] where:
- [tex]\( a = 4.9 \)[/tex],
- [tex]\( b = 3 \)[/tex],
- [tex]\( c = 20 \)[/tex].
The quadratic formula is given by:
[tex]\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, compute the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values:
[tex]\[ \Delta = 3^2 - 4 \cdot 4.9 \cdot 20 \][/tex]
[tex]\[ \Delta = 9 - 392 \][/tex]
[tex]\[ \Delta = -383 \][/tex]
Since the discriminant is negative, we conclude:
1. The quadratic equation [tex]\( 4.9 t^2 + 3 t + 20 = 0 \)[/tex] has no real roots.
2. Therefore, there is no real time [tex]\( t \)[/tex] when the divers will reach the water’s surface based on the given conditions.
Conclusively, based on the results, the divers do not reach the water with the given initial conditions.
