A dart gun where you compress the spring and it shoots a rubber dart of a mass 0.122 kg. The dart is shot vertically and gains 2.50 m.

a) what speed did the dart leave the toy gun?
b) If the spring is compressed 0.125 m what is the elasticity coefficient of the spring?

Initial velocity=√(2gh)

h=height reached.

Spring constant=k
compression = x = 0.125m
Potential energy=(1/2)mv²=mgh
=(1/2)kx²
solve for k in
(1/2)kx² = mgh
k=2gh/x²

What happen to the m? It cancelled out?

Yes, but only in the first part.

m was inadvertently omitted in the second part, my apologies.
k=2mgh/x²

Thank you!

You're welcome! :)

To solve these problems, we can use the principles of energy conservation and Hooke's Law. The first step is to calculate the potential energy stored in the spring when it is compressed. Let's denote the spring constant as "k" and the compression distance as "x."

a) To find the speed at which the dart leaves the toy gun, we need to apply the principle of conservation of mechanical energy. At the moment the dart is released, all of the potential energy stored in the compressed spring will be converted into kinetic energy. The formula for potential energy stored in a spring is given by:

Potential energy (PE) = 0.5 * k * x^2

Given that the spring is compressed by 0.125 m, which means x = 0.125 m, we need to find the spring constant "k" to calculate the potential energy.

b) Hooke's Law states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed. Mathematically, it can be expressed as:

Force (F) = -k * x

Where "F" is the force, "k" is the spring constant, and "x" is the compression or stretch distance.

To calculate the spring constant "k," we rearrange the equation:

k = -F / x

Given the mass of the dart (m = 0.122 kg) and the distance the dart rises (h = 2.50 m), we can find the force exerted by gravity using Newton's second law:

Force (F) = m * g

where "g" is the acceleration due to gravity (approximately 9.8 m/s^2).

Let's calculate the answers step by step:

a) To find the speed of the dart when it leaves the toy gun, we need to equate potential energy to kinetic energy:

Potential Energy = Kinetic Energy

0.5 * k * x^2 = 0.5 * m * v^2

Rearranging the equation and plugging in the known values:

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

b) To find the spring constant "k," we use the equation:

k = -F / x

where "F" is calculated using:

F = m * g

Then, substitute the known values to calculate the spring constant:

k = -(m * g) / x

Now, let's plug the given values into the equations and solve for the desired quantities.