At an amusement park, a swimmer uses a water slide to enter the main pool. The swimmer slides without friction, and descends through a vertical height of 2.98m.

Find the swimmer's speed at the bottom of the slide if she starts with an initial speed of 0.750 m/s.

To find the swimmer's speed at the bottom of the slide, we can use the principle of conservation of mechanical energy. The total mechanical energy of the swimmer at the top of the slide will be equal to the total mechanical energy at the bottom of the slide, neglecting any losses due to friction or air resistance.

The total mechanical energy (E) is the sum of the kinetic energy (KE) and the potential energy (PE):

E = KE + PE

At the top of the slide, the swimmer only has kinetic energy:

E_top = KE_top = 0.5 * m * v_top^2

where m is the mass of the swimmer and v_top is the initial speed of the swimmer at the top of the slide.

At the bottom of the slide, the swimmer will have both kinetic energy and potential energy:

E_bottom = KE_bottom + PE_bottom

The potential energy at the bottom of the slide is given by:

PE_bottom = m * g * h

where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height of the slide.

Since there is no friction, the mechanical energy is conserved, so we can equate E_top and E_bottom:

E_top = E_bottom

0.5 * m * v_top^2 = KE_bottom + PE_bottom

Substituting the expressions for PE_bottom and KE_bottom:

0.5 * m * v_top^2 = m * g * h + 0.5 * m * v_bottom^2

where v_bottom is the speed of the swimmer at the bottom of the slide.

Now, we can rearrange the equation to solve for v_bottom:

0.5 * m * v_top^2 - 0.5 * m * v_bottom^2 = m * g * h

Simplifying:

0.5 * m * (v_top^2 - v_bottom^2) = m * g * h

Dividing both sides by m and canceling common terms:

0.5 * (v_top^2 - v_bottom^2) = g * h

Rearranging the equation to solve for v_bottom:

v_bottom^2 = v_top^2 - 2 * g * h

Finally, taking the square root of both sides, we can find the swimmer's speed at the bottom of the slide:

v_bottom = sqrt(v_top^2 - 2 * g * h)

Plugging in the values given in the question, we can calculate the speed:

v_bottom = sqrt(0.750^2 - 2 * 9.8 * 2.98)

v_bottom ≈ sqrt(0.5625 - 58.164)

v_bottom ≈ sqrt(-57.6015)

Since the square root of a negative number is not defined in the real number system, it means that the swimmer does not have enough initial speed to reach the bottom of the slide. Therefore, the speed at the bottom of the slide is 0 m/s.