To enter the main pool at an amusement part, a swimmer uses a water slide which has a vertical height of 2.65 m. Find her speed at the bottom of the slide if she starts with an initial speed of 0.950 m/s.
To find the speed of the swimmer at the bottom of the slide, we can use the principles of conservation of energy. We need to consider the initial kinetic energy and the change in potential energy.
Here's how you can calculate it step by step:
1. Determine the initial potential energy (PE_initial) of the swimmer at the top of the slide using the formula:
PE_initial = m * g * h
where m is the mass of the swimmer, g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the slide (2.65 m).
2. Calculate the initial kinetic energy (KE_initial) of the swimmer at the top of the slide using the formula:
KE_initial = 0.5 * m * v_initial^2
where m is the mass of the swimmer and v_initial is the initial speed (0.950 m/s).
3. Since energy is conserved, the initial potential energy will be converted into final kinetic energy (KE_final) at the bottom of the slide. Therefore, we can equate the initial potential energy (PE_initial) to the final kinetic energy (KE_final).
PE_initial = KE_final
4. Rearrange the equation to solve for the final velocity (v_final):
KE_final = 0.5 * m * v_final^2
v_final^2 = (2 * PE_initial) / m
v_final = √((2 * PE_initial) / m)
5. Substitute the known values into the equation. Assuming a mass of 70 kg for the swimmer, the calculation becomes:
v_final = √((2 * 70 * 9.8 * 2.65) / 70)
6. Simplify the equation to find the final velocity:
v_final = √(2 * 9.8 * 2.65)
v_final = √(51.96)
v_final ≈ 7.21 m/s
Therefore, the swimmer will have a speed of approximately 7.21 m/s at the bottom of the slide.