A slingshot with k = 400. N/m is pulled back 0.0500 m to shoot a 0.100-kg stone. If the slingshot projects the stone straight up in the air, (a) what is the speed when the stone is first released, and (b) what is the maximum height to which the stone will rise? [neglect friction]

To solve this problem, we will use the principle of conservation of mechanical energy.

(a) To find the speed when the stone is first released, we can use the equation for potential energy stored in a spring:

PE = (1/2) k x^2

where PE is the potential energy, k is the spring constant, and x is the displacement of the spring (0.0500 m in this case).

The potential energy stored in the spring is converted into the kinetic energy of the stone when it is released. Therefore, we can equate the potential energy to the kinetic energy:

PE = KE

(1/2) k x^2 = (1/2) m v^2

where m is the mass of the stone (0.100 kg in this case) and v is the velocity we are trying to find.

Rearrange the equation to solve for v:

v^2 = (k x^2) / m

v = sqrt((k x^2) / m)

Now we can substitute the given values into the equation to find v:

v = sqrt((400 N/m) * (0.0500 m)^2 / 0.100 kg)
= sqrt(100 N * m)
= 10 m/s

Therefore, the speed when the stone is first released is 10 m/s.

(b) To find the maximum height to which the stone will rise, we can use the conservation of mechanical energy again.

At the maximum height, the kinetic energy of the stone will be zero, since the stone momentarily stops moving before it starts falling back down. Therefore, the potential energy at the maximum height should equal the initial potential energy stored in the spring.

PE_max_height = (1/2) k x^2

PE_max_height = (1/2) (400 N/m) (0.0500 m)^2
= 0.500 J

Using the conservation of energy:

PE_initial = PE_max_height + KE_max_height

Since the stone comes to rest at the maximum height, KE_max_height is zero.

(1/2) k x^2 = PE_max_height
(1/2) (400 N/m) (0.0500 m)^2 = 0.500 J

Solving for x:

x = sqrt((2 PE_max_height) / k)
= sqrt((2 * 0.500 J) / 400 N/m)
= sqrt(0.00250 m^2/N)
= 0.0500 m

Therefore, the maximum height to which the stone will rise is 0.0500 m.

In summary, the speed when the stone is first released is 10 m/s, and the maximum height to which the stone will rise is 0.0500 m.