To solve these problems, we can approach each question in steps, starting with the speed of the ball as it touches the spring, followed by determining the force constant of the spring.

1. Speed of the Ball Just Before Touching the Spring

Given:

• Mass, $m = 1.63 \, \text{kg}$
• Initial height, $h = 61.2 \, \text{cm} = 0.612 \, \text{m}$
• Gravitational acceleration, $g = 9.81 \, \text{m/s}^2$

When the ball is released from height $h$, it falls under gravity. The potential energy at height $h$ converts into kinetic energy just before it touches the spring.

Solve Symbolically

Using energy conservation:

$mgh = \frac{1}{2}mv^2$

Solving for $v$:

$v = \sqrt{2gh}$

Solve Numerically

Substitute $g = 9.81 \, \text{m/s}^2$ and $h = 0.612 \, \text{m}$:

$v = \sqrt{2 \cdot 9.81 \cdot 0.612}$

$v \approx 3.47 \, \text{m/s}$

So, the speed of the ball just as it touches the spring is approximately: $v = 3.47 \, \text{m/s}$

2. Force Constant of the Spring

When the ball compresses the spring by a distance $d = 7.62 \, \text{cm} = 0.0762 \, \text{m}$, the kinetic energy just before contact with the spring is converted into elastic potential energy in the spring.

Solve Symbolically with Variables $m, g, h,$ and $d$

Using the conservation of energy between the initial height and the maximum compression:

$mgh = \frac{1}{2}kd^2$

Solving for $k$:

$k = \frac{2mgh}{d^2}$

Solve Symbolically with Variables $m, g, v,$ and $d$

Alternatively, using the speed $v$ of the ball just before contact:

$\frac{1}{2}mv^2 = \frac{1}{2}kd^2$

Solving for $k$:

$k = \frac{mv^2}{d^2}$

Solve Numerically

Using $m = 1.63 \, \text{kg}$, $g = 9.81 \, \text{m/s}^2$, $h = 0.612 \, \text{m}$, and $d = 0.0762 \, \text{m}$:

$k = \frac{2 \cdot 1.63 \cdot 9.81 \cdot 0.612}{(0.0762)^2}$

$k \approx 2725.46 \, \text{N/m}$

So, the force constant of the spring is approximately: $k = 2725.46 \, \text{N/m}$

Summary of Answers:

1. Speed of the ball just before it touches the spring: $v = 3.47 \, \text{m/s}$

2. Force constant of the spring: $k = 2725.46 \, \text{N/m}$

Would you like more detailed explanations or have further questions?

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Related Questions:

1. How would the speed change if the height was doubled?
2. What would happen to the force constant if the compression distance was halved?
3. How does gravitational acceleration affect the speed calculation?
4. What additional forces might we consider if energy loss wasn't neglected?
5. How would the system behave if the spring were not massless?

Tip:

Always convert all units to the SI system (meters, kilograms, seconds) before performing calculations to ensure consistency.