Question

A dart gun contains a spring of spring constant 11 N/m which is used to fire a dart of mass. The dart leaves the gun at a speed of 5.9 m/s after the spring is compressed 1 cm. 1. What is the weight in newtons of the dart?2. What is the dart speed when it hits the floor in m/s if it is fired horizontally at a height of 2 meters?

Answer 1

1. 0.00031 N

2. 8.6 m/s

Explanation:

Given:

Spring constant (k) = 11 N/m

Speed (v) = 4.7 m/s

x = 1 cm = 0.01 m

1.

We can determine the mass by conservation of energy and then calculate the weight in the following way:

`\begin{gathered} (1)/(2)kx^2=(1)/(2)mv^2 \\ \\ \text{ We replacing} \\ \\ 11\cdot \left(0.01\right)^2=m\cdot \left(5.9\right)^2\: \\ \\ m=(11\cdot(0.01)^2)/((5.9)^2) \\ \\ m=3.16\cdot10^(-5)\text{ kg} \\ \\ W=m\cdot g \\ \\ W=3.16\cdot10^(-5)\cdot9.8 \\ \\ W=0.00031\text{ N} \end{gathered}`

2.

The speed can also be determined by the conservation of energy, thus:

`\begin{gathered} (1)/(2)mu^2=mgh+(1)/(2)mv^2 \\ \\ \text{ We relaplacing:} \\ \\ (1)/(2)u^2=(9.8)(2)+(1)/(2)(5.9)^2 \\ \\ u^2=2\cdot((9.8)(2)+(1)/(2)(5.9)^2) \\ \\ u=√(74.01) \\ \\ u=8.6\text{ m/s} \end{gathered}`