A particle, starting from point A in the drawing at a height h0 = 3.2 m, is projected down the curved runway. Upon leaving the runway at point B, the particle is traveling straight upward and reaches a height hf = 4.1 m above the floor before falling back down. Ignoring friction and air resistance, find the speed of the particle at point A.

To find the speed of the particle at point A, we can use the principle of conservation of energy.

The initial potential energy of the particle at point A is given by mgh0, where m is its mass and g is the acceleration due to gravity. The final potential energy of the particle at point B is given by mghf.

According to the conservation of energy, the initial potential energy at point A is equal to the final kinetic energy at point B.

The initial potential energy at point A is converted into kinetic energy at point B. The kinetic energy can be calculated using the formula: (1/2)mv^2, where v is the speed of the particle at point A.

So we have:

mgh0 = (1/2)mv^2

We can cancel the mass m from both sides of the equation:

gh0 = (1/2)v^2

Now we can rearrange the equation to solve for v:

v^2 = 2gh0

Taking the square root of both sides gives us:

v = sqrt(2gh0)

Substituting the given values:

v = sqrt(2 * 9.8 * 3.2)

Calculating this expression will give you the speed of the particle at point A.