At an amusement park, a swimmer uses a water slide to enter the main pool. If the swimmer slides without friction, and descends through a vertical height of 3.01 m, and has a initial speed of 0.970 m/s, what is her speed at the bottom of the slide?
??? m/s
To determine the swimmer's speed at the bottom of the slide, we can make use of the principle of conservation of energy.
At the top of the slide, the swimmer has potential energy (due to her height) and kinetic energy (due to her initial speed). As she slides down, her potential energy is converted into kinetic energy. Assuming no energy is lost due to friction, the total mechanical energy (potential energy + kinetic energy) remains constant throughout the slide.
First, let's find the potential energy (PE) and kinetic energy (KE) at the top of the slide:
PE = m * g * h
Where m is the mass of the swimmer, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height (3.01 m).
PE = m * 9.8 * 3.01
Next, we find the kinetic energy (KE) at the top of the slide:
KE = 0.5 * m * v²
Where v is the initial speed of the swimmer (0.970 m/s).
KE = 0.5 * m * (0.970)²
Since the total mechanical energy is conserved, we can set the potential energy at the top equal to the kinetic energy at the bottom of the slide:
PE = KE
m * 9.8 * 3.01 = 0.5 * m * v²
Now, solve the equation for v²:
v² = (m * 9.8 * 3.01) / (0.5 * m)
The mass of the swimmer cancels out:
v² = 9.8 * 3.01 / 0.5
Calculate the value:
v² = 58.2476
Finally, take the square root of both sides to find the speed at the bottom of the slide:
v = √58.2476
v ≈ 7.63 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 7.63 m/s.
To find the swimmer's speed at the bottom of the slide, we can use the principle of conservation of energy. At the top of the slide, the swimmer has gravitational potential energy due to her vertical height above the ground. At the bottom of the slide, this potential energy is converted into kinetic energy (energy of motion).
The equation for gravitational potential energy is given by:
PE = m * g * h
Where:
PE is the potential energy
m is the mass of the object (which cancels out if we assume the swimmer's mass is constant)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the vertical height
The equation for kinetic energy is given by:
KE = (1/2) * m * v^2
Where:
KE is the kinetic energy
m is the mass of the object (which cancels out if we assume the swimmer's mass is constant)
v is the velocity
Based on the conservation of energy principle, we can equate the potential energy at the top to the kinetic energy at the bottom:
m * g * h = (1/2) * m * v^2
Cancelling out the mass, we get:
g * h = (1/2) * v^2
Now we can solve for v:
v^2 = 2 * g * h
Taking the square root of both sides, we have:
v = sqrt(2 * g * h)
Substituting the given values:
g = 9.8 m/s^2
h = 3.01 m
v = sqrt(2 * 9.8 * 3.01) = 7.38 m/s
Therefore, the swimmer's speed at the bottom of the slide is approximately 7.38 m/s.