# A block is held 1.3m above a spring and is dropped. The spring compresses 6cm before sending the...

## Question:

A block is held 1.3m above a spring and is dropped. The spring compresses 6cm before sending the ball into the air.

How fast is the ball going when it hits the spring?

What is the spring constant?

How high in the air does the ball go after hitting the spring?

## Energy conservation

The conservation of energy states that there won't be any change in the total energy for a conservative system. Energy gets transferred from one form to another. If no dissipative forces are acting in the system such as frictional force, then the system is said to be conservative.

Step 1: Determining speed

To determine the speed we can use the conservation of energy from potential energy to kinetic energy.

Therefore,

{eq}\begin{align*} &mgh=\frac{1}{2}mv^2\\ \Rightarrow & v = \sqrt{2gh} \end{align*} {/eq}

Putting the values in the above equation we get,

{eq}\begin{align*} &v = \sqrt{2\times 9.8 \times 1.3}\\ \Rightarrow & v \approx 5.05\ m/s \end{align*} {/eq}

Step 2: Determining spring constant

To determine the spring constant we can use the energy conservation between kinetic energy and the potential energy stored in the spring.

Therefore,

{eq}\begin{align*} & \frac{1}{2}kx^2=\frac{1}{2}mv^2\\ \Rightarrow & k = \frac{mv^2}{x^2}\\ \end{align*} {/eq}

Putting the values in the above equation we get,

{eq}\begin{align*} &k = \frac{m\times 25.48}{0.06^2}\\ \Rightarrow & k = 7077.8m\ kg/s^2 \end{align*} {/eq}

Step 3: Determining the height after hitting the spring

As there is no loss of energy, the ball will rebound to the same height as it was before. Hence, the height it will reach after hitting will be 1.3 m.

Conclusion

Velocity of the ball will be {eq}\begin{align*} 5.05 \ m/s \end{align*} {/eq}.

Spring constant of the spring is {eq}\begin{align*} k = 7077.8m\ kg/s^2 \end{align*} {/eq} where 'm' is the mass of the ball.

The height the ball will reach after hitting the spring will be {eq}\begin{align*} 1.3\ m \end{align*} {/eq}.

Note: The spring constant is in terms of the mass because the mass of the ball is unknown in this problem.