A person on a trampoline bounces straight upward with an initial speed of 4.7m/s . What is the magnitude of the person's speed when she returns to her initial height?

d = (Vf^2-Vo^2) / 2g,

d = (0-(4.7)^2) / 19.6 = 1.13m,up.

Vf^2 = Vo^2 + 2gd,
Vf^2 = 0 + 2*9.8*1.13 = 22.15,

Vf = 4.7m/s.

Well, when the person reaches their highest point, their speed will be equal to zero. But let's not worry about their grumpy return to the ground just yet – let's appreciate their momentary weightlessness!

To determine the magnitude of the person's speed when she returns to her initial height, we need to consider the conservation of mechanical energy.

When the person reaches their maximum height, all of their initial kinetic energy is converted into potential energy. This means that the gravitational potential energy at the maximum height is equal to the initial kinetic energy.

At the maximum height, the potential energy can be expressed as:

PE = mgh

Where "m" is the mass of the person, "g" is the acceleration due to gravity (approximately 9.8 m/s^2), and "h" is the maximum height reached.

At the maximum height, the kinetic energy is zero since the person momentarily stops. Therefore, we have:

PE = KE_initial

mgh = (1/2)mv^2_initial

Simplifying and rearranging the equation, we get:

v^2_initial = 2gh

Now, let's solve for the magnitude of the person's speed when she returns to her initial height:

v_final = √(2gh)

Given:
v_initial = 4.7 m/s
g = 9.8 m/s^2

Substituting the values, we have:

v_final = √(2 * 9.8 * h)

Since the person's initial speed is equal to their final speed, we can rewrite the equation as:

v_initial = √(2 * 9.8 * h)

Solving for "h", we have:

h = (v_initial^2) / (2 * 9.8)

Plugging in the given value for v_initial:

h = (4.7^2) / (2 * 9.8)

Calculating the value:

h ≈ 1.1025 m

Therefore, the magnitude of the person's speed when she returns to her initial height is approximately 4.7 m/s.

To find the magnitude of the person's speed when she returns to her initial height, we can use the principle of conservation of mechanical energy.

The total mechanical energy of the person-trampoline system remains constant throughout the motion. At the highest point of the motion, all of the initial kinetic energy is converted to potential energy. When the person returns to the initial height, all of the potential energy is converted back to kinetic energy.

Therefore, we can equate the initial kinetic energy to the final kinetic energy. The formula for kinetic energy is:

K = (1/2) * m * v^2,

where K is the kinetic energy, m is the mass of the person, and v is the velocity.

Since we are given the initial speed (v), we can calculate the initial kinetic energy using the formula.

Initial kinetic energy (K_i) = (1/2) * m * v^2

To find the final speed when the person returns to the initial height, we need to know the mass of the person.

Once we know the mass, we can equate the initial kinetic energy (K_i) to the final kinetic energy (K_f):

K_i = K_f

(1/2) * m * v^2 = (1/2) * m * v_f^2

Since the person returns to the same height, the potential energy is zero at both the initial and final positions. We can use this fact to rewrite the equation as:

K_i - 0 = K_f - 0

(1/2) * m * v^2 = (1/2) * m * v_f^2

We can divide both sides of the equation by (1/2) * m:

v^2 = v_f^2

Taking the square root of both sides, we get:

v = v_f

Therefore, the magnitude of the person's speed when she returns to her initial height is equal to the magnitude of the initial speed, which is 4.7 m/s.