A child wants to throw a rock into a puddle which is 20m away and 20m down a hill. If he can throw the rock at ana ngle of 60, what velocity does he need to release the rock at?

To determine the velocity the child needs to release the rock at, we can use the concepts of projectile motion and trigonometry. The velocity of the rock can be broken down into its horizontal and vertical components.

First, let's find the horizontal component of the velocity. We know that the horizontal distance to the puddle is 20m, and we can use the equation:

horizontal distance = horizontal velocity x time

Since there is no horizontal acceleration (assuming no air resistance), we can set the horizontal velocity equal to the initial velocity (V₀) multiplied by the cosine of the launch angle:

20m = V₀ * cos(60°) * time

Next, let's find the vertical component of the velocity. We know that the vertical displacement is 20m. Again, using the equation of motion for vertical displacement:

vertical displacement = (vertical velocity x time) + (0.5 x acceleration x time²)

Since the rock is thrown vertically, the initial vertical velocity is equal to the initial velocity (V₀) multiplied by the sine of the launch angle:

20m = (V₀ * sin(60°) * time) + (0.5 x (-9.8m/s²) x time²)

Now we have a system of two equations with two unknowns (V₀ and time). We can solve these equations simultaneously to obtain the values.

To calculate the velocity required to throw the rock into the puddle, you can follow these steps:

1. Solve the first equation for time using the given horizontal distance of 20m and the launch angle of 60°, which gives us the horizontal time.
2. Substitute this value of time into the second equation, along with the known vertical displacement of 20m and the acceleration due to gravity (-9.8m/s²), and solve for V₀.
3. The resulting value of V₀ will be the velocity required to release the rock at.

Note: In the equation, it is assumed that there is no air resistance, the ground level is at the same height as the starting point (no elevation change), and the rock is thrown horizontally.

By following these steps and applying the appropriate mathematical calculations, you can determine the velocity needed to release the rock.