a toy spring gun is used to shoot 28-g bullets vertically upward. the un deformed length is 150 mm, it is compressed to a length of 25 mm when the gun is ready to be shoot and expands to a length of 75mm as the bullet leaves the gun. a force of 45 N is required to maintain the spring in firing position when the length of the spring is 25mm. determine the velocity of the bullet as leaves the gun and the maximum height reached by a bullet
To determine the velocity of the bullet as it leaves the gun and the maximum height reached, we can use the principles of energy conservation.
First, let's find the spring constant (k) of the toy gun. The spring constant relates the force required to compress or extend a spring to the displacement:
k = F / x
where F is the force (45N) and x is the displacement (25mm = 0.025m).
k = 45N / 0.025m
k = 1800 N/m
Next, let's calculate the potential energy stored in the spring when it is fully compressed to a length of 25mm:
PE = (1/2) k x^2
PE = (1/2) * 1800 N/m * (0.025m)^2
PE = (1/2) * 1800 N/m * 0.000625 m^2
PE = 0.5625 J
This potential energy is converted into the kinetic energy of the bullet as it leaves the gun:
KE = (1/2) mv^2
where m is the mass of the bullet (28g = 0.028kg), and v is the velocity we want to determine.
Now, we equate PE and KE:
0.5625 J = (1/2) * 0.028kg * v^2
Solving for v:
v^2 = (2 * 0.5625 J) / 0.028kg
v^2 = 40.1786 m^2/s^2
v = sqrt(40.1786 m^2/s^2)
v ≈ 6.34 m/s
Therefore, the velocity of the bullet as it leaves the gun is approximately 6.34 m/s.
To find the maximum height reached by the bullet, we can use the equation of motion:
v^2 = u^2 + 2as
where u is the initial velocity (6.34 m/s), a is the acceleration (in this case, acceleration due to gravity, -9.8 m/s^2), and s is the displacement.
Since the bullet goes upward, it will eventually come to a stop at the maximum height, so the final velocity (v) will be zero. Rearranging the equation:
0 = u^2 + 2as
0 = (6.34 m/s)^2 + 2(-9.8 m/s^2)s
0 = 40.1796 m^2/s^2 - 19.6 m/s^2 s
19.6 m/s^2 s = 40.1796 m^2/s^2
s = 40.1796 m^2/s^2 / 19.6 m/s^2
s ≈ 2.05 m
Therefore, the maximum height reached by the bullet is approximately 2.05 meters.