"The launching mechanism of a toy gun consists of a spring of unknown spring constant. If the spring is compressed a distance of 0.139 m and the gun is fired vertically, the gun can launch a 25.1g projectile from rest to a maximum height of 24.5 m above the starting point of the projectile. Neglecting all resistive forces, determine the spring constant."© BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf
SOLUTION This solution is FREE courtesy of BrainMass!
If you compress a spring to some extent, the work done by your body will be stored in the spring as the potential energy of the spring. When the spring is released, the energy is also released as some other form.
The potential energy of the spring PE = (1/2)*k * (x-x0)^2
where k is the force constant and (x-x0) is the compresion. Thus in our case, the energy stored is
PE = 0.5 * k * 0.139 = 0.0695 k Joules ......(1)
This much energy will appear as the kinetic energy of the projectile when it is released.
we have KE = (1/2)mV^2 .....(2)
we have mass, but how to calculate the velocity ?
ok, suppose we project a body with a KE, then it can go up to a height where its KE is exhausted. We cannot say that it is lost but it will get stored in the body as its potential energy.
At the highest point, the body just stops and at this position we will write for the body,
KE(initial) = PE (final)
(1/2)mV^2 = mgH or from this V^2 = 2gH
Now the velocity in terms of the maximum height is V = sqrt(2gH)
We will plug this equation in to our second equation to get the KE of the projected body.
KE = (1/2)m (V^2 ) = (1/2)m* 2gH .....(3)
Now we will equate both 1 and 3, since they are the same amount of energy (though in separate forms)
(1/2)m* 2gH = 0.0695 k
from this, K = (1/2)m* 2gH/0.069
= 0.5 * 25.1 * 10^-3kg * 9.8m/s^2 * 24.5m/0.069
Suppose I don't know the unit of K, how can I find it ?
we have F = k x
I know the units of F and X as Newton and meter respectively
thus K = F/x = Newton/meter
Thus our answer is k = 43.67 N/m© BrainMass Inc. brainmass.com December 24, 2021, 4:49 pm ad1c9bdddf>