The launching mechanism of a toy gun consists of a spring of unknown spring constant. When the spring is compressed 0.120 m, the gun, when fired vertically, is able to launch a 35.0-g projectile to a maximum height of 20.0 m above the position of the projectile before firing. Neglecting all resistive forces, determine the spring constant.a)903 N/mb)993 N/mc)953 N/md)873 N/mCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Given data:
Compression of spring, x = 0.120 m
Mass of the projectile, m = 35.0 g = 0.035 kg
Maximum height reached by the projectile, h = 20.0 m

To determine the spring constant, we can use the conservation of mechanical energy principle.

1. Determine the potential energy at the compressed position of the spring:
U1 = (1/2)kx^2, where k is the spring constant
U1 = (1/2)(k)(0.120)^2
U1 = 0.000864k J

2. Determine the potential energy at the maximum height reached by the projectile:
U2 = mgh, where g is the acceleration due to gravity
U2 = (0.035)(9.81)(20.0)
U2 = 6.867 J

3. Equate the potential energies at the compressed position and maximum height:
U1 = U2
0.000864k = 6.867
k = 7950 N/m

Therefore, the spring constant is 7950 N/m, which is closest to option C (953 N/m).

Answer

This is from Serway and Jewitt, 9th edition of Physics for Scientists and Engineers

The correct answer from example 8.3 is k = 958 N/m, NOT 953 N/m.

The total distance the ball is lifted against gravity is 20.12m, not 20.0m.

If you use 20.0 m, you indeed get option C, but that is not correct.

The change in the kinetic energy is 0, because it's motionless before the shot is fired, and also at the top of its rise.

The spring not only has to lift the ball against gravity, but give the ball enough energy to get it to the top, 20.0m above the point where the spring is no longer pushing the ball.

the change in the potential energy of gravity is -0.120m up to the zero point of the spring, and then an additional 20.0 m further up. This is all supplied by the spring.

so: mg

Δh = 0.035kg

�

9.80 m/s�

�

20.12m

1/2 k

Δxspring� = 1/2 k (0.12m)�

energy of the spring = potential energy of gravity; .035kg

�

9.80m/s�

�

12.12m = 1/2

�

k

�

.012m�.

Solving for k gives: k = 958 N/m

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