In a bungee jump, Henry falls while attached to an elastic cord - Leaving Cert Physics - Question 6 - 2017

To derive the equation of motion, we start with the basic definitions of acceleration:

1. Acceleration ($a$) is defined as the change in velocity ($v - u$) over time ($t$): $a = \frac{v - u}{t}$

2. Rearranging gives us: $v = u + at$

3. The displacement ($s$) can be written as: $s = ut + \frac{1}{2}at^2$

4. By substituting $t = \frac{v - u}{a}$ into the displacement equation, we can derive: $s = u \left(\frac{v - u}{a}\right) + \frac{1}{2}a \left(\frac{v - u}{a}\right)^2$

5. Simplifying this gives the relation: $v^2 = u^2 + 2as$

Using the derived equation, we can find the speed ($v$) after falling 16 m:

1. Given:

   • Initial velocity ($u$) = 0
   • Acceleration ($a$) = 9.8 m/s$^2$
   • Displacement ($s$) = 16 m

2. Plugging into the formula: $v^2 = u^2 + 2as = 0 + 2(9.8)(16)$ $v^2 = 313.6$ $v = \sqrt{313.6} = 17.7 \, \text{m/s}$

Hooke's law states that the restoring force exerted by a spring or elastic cord is directly proportional to the distance it is stretched or compressed. Mathematically, this can be represented as:

$F = -kx$

where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement from the equilibrium position.

Simple harmonic motion (SHM) is a type of periodic motion in which an object moves back and forth around an equilibrium position. The motion is characterized by an acceleration that is directly proportional to the displacement from the equilibrium position and is always directed towards that position. Mathematically, it can be expressed as:

$a = -\omega^2 x$

where eta is angular frequency and $x$ is the displacement.

Using Hooke's law, we can find the length of the cord at rest when Henry is suspended:

1. The force due to Henry's weight is: $F = mg = 60 imes 9.8 = 588 \, \text{N}$

2. From Hooke's law:

   $$$$

ightarrow 588 = 250x$$

3. Rearranging gives us: $x = \frac{588}{250} = 2.352 \, \text{m}$

4. Total length of the cord when suspended at rest is: $L = 32 + x = 32 + 2.352 = 34.352 \, \text{m}$

(i) The maximum acceleration in simple harmonic motion can be calculated as:

1. Using the formula for maximum acceleration: $a_{max} = \omega^2 A$ where $A$ is the amplitude (maximum displacement)

2. Given:

   • Maximum displacement (A) = 1.2 m
   • Calculating angular frequency ($\omega$): Given that $k = 250 \, \text{N/m}$ and $m = 60 \, \text{kg}$, $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{250}{60}}\approx 2.02 \, \text{rad/s}$

3. Thus: $a_{max} = (2.02)^2(1.2) \approx 4.9 \, \text{m/s}^2$

(ii) The period of oscillation ($T$) for simple harmonic motion can be calculated using:

$T = \frac{2\pi}{\omega}$

1. Calculating $T$: $T = \frac{2\pi}{2.02} \approx 3.10 \, \text{s}$

1. In the diagram:

   • Draw an arrow pointing downward for the gravitational force (weight) acting on Henry.
   • Draw a shorter arrow pointing upward to represent the elastic force exerted by the cord.

2. Label the forces appropriately to show that the downward force (mg) is greater than the upward force (kx) at the lowest point.