A spring-loaded toy dart is used to shoot a dart straight up in the air, and the dart reaches a maximum height of 16m. The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the dart go this time?

To find out how far up the dart goes the second time, let's use the concept of conservation of energy.

The potential energy of the dart at its maximum height can be calculated using the formula:

Potential Energy (PE) = mass (m) × gravitational acceleration (g) × height (h)

Since the mass of the dart remains constant and gravitational acceleration (g) is constant, we can say that the potential energy of the dart is directly proportional to the height (h).

Let's denote the initial compressed distance of the spring before firing as x.

In the first case, the dart reaches a maximum height (h1) of 16m.

In the second case, when the spring is compressed only half as far, the new compressed distance is x/2.

Using the principle of conservation of energy, we can write the equation:

Potential Energy 1 = Potential Energy 2

mgh1 = mgh2

Since the mass (m) is common on both sides, it cancels out.

gh1 = gh2

We can simplify this equation by dividing both sides by g:

h1 = h2

So, the maximum height (h2) the dart will reach the second time will be the same as the maximum height (h1) in the first case, which is 16m.

Therefore, the dart will reach a height of 16m the second time, regardless of the spring compression.