Question

1.A dwarf drops from a tree and when it hits the ground it crouched down a distance of 0.4 m. If the dwarf dropped down from a height of 3.0 m what was the average force it absorbed over the crouching distance of 0.4 m? (Please show any sigfigs and any equations you've used) thank you kindly.

2. Initially the goblin and the dwarf have the same velocity, but the dwarf has asthma and begins to slow down. He slows his pace at a constant rate of -0.6m/s^2 but the goblin maintains his velocity at 8 m/s. If the distance is 15m between the dwarf and the goblin when the dwarf began to slow down how long would it take for the dwarf to catch up? (apparently there are 3 parts to solving this problem.. please explain.)

3. The goblins fists cut through the air and connects with the dwarfs face in an elastic collision. The goblins fists each weigh 2kg and the dwarfs head weighs 5kg. What is the kinetic energy imparted to the dwarfs face by the goblins fists if the goblins fists had an initial velocity of 5m/s?

Much help would be greatly appreciated as I am stuck on this problem. Please show any steps and equations used to solve this problem, thank you.

Answer 1

Answer on Question#38525- Physics – Other

1. A dwarf drops from a tree and when it hits the ground it crouched down a distance of $0.4\,\mathrm{m}$. If the dwarf dropped down from a height of $3.0\,\mathrm{m}$ what was the average force it absorbed over the crouching distance of $0.4\,\mathrm{m}$?

Solution.

$$\begin{array}{l} m = x\, kg, \\ h = 3\, m, \\ d = 0.4\, m, \\ g = 9.8\, m / s^2. \end{array}$$

Potential energy of dwarf:

$$E = mgh$$

Work to overcome the force

$$W = Fd$$

Potential energy is equal to work

$$\begin{array}{l} E = W \\ mgh = Fd \\ F = mg\, \frac{h}{d} \end{array}$$

You must have a mass of dwarf $m$

Answer.

$$F = mg\, \frac{h}{d} = m \cdot 9.8 \cdot \frac{3}{0.4}$$

2. Initially the goblin and the dwarf have the same velocity, but the dwarf has asthma and begins to slow down. He slows his pace at a constant rate of $-0.6\,\mathrm{m/s^2}$ but the goblin maintains his velocity at $8\,\mathrm{m/s}$. If the distance is $15\,\mathrm{m}$ between the dwarf and the goblin when the dwarf began to slow down how long would it take for the dwarf to catch up?

Solution.

$$\begin{array}{l} v = 8\,\mathrm{m/s} - \text{goblin's velocity}, \\ a = 0.6\,\mathrm{m/s^2} - \text{dwarf's acceleration} \\ x_0 = 15\,\mathrm{m}. \end{array}$$

Equations of motion:

for goblin: $x = vt = 8t$

for dwarf: $x = x_0 + vt + \frac{at^2}{2} = 15 + 8t - 0.3t^2$

The ways of dwarf and goblin are equal

$$\begin{array}{l} 8t = 15 + 8t - 0.3t^2 \\ 0 = 15 - 0.3t^2 \\ t = \sqrt{\frac{2x_0}{-a}} = \sqrt{\frac{2 \cdot 15}{0.6}} = \sqrt{50} = 7.071\,\mathrm{s}. \end{array}$$

Answer.

$$t = 7.071\,\mathrm{s}.$$

3. The goblins fists cut through the air and connects with the dwarfs face in an elastic collision. The goblins fists each weigh $2\mathrm{kg}$ and the dwarfs head weighs $5\mathrm{kg}$. What is the kinetic energy imparted to the dwarfs face by the goblins fists if the goblins fists had an initial velocity of $5\mathrm{m/s}$?

Solution.

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter.

$\mathrm{m}_1 = 2\mathrm{kg} - \mathrm{mass}$ of goblins fist,

$\mathrm{m}_2 = 5\mathrm{kg} - \mathrm{mass}$ of dwarfs head,

$\mathrm{v}_0 = 5\mathrm{m/s} - \mathrm{initial~velocity}$ of the goblins fists.

The conservation of the total momentum demands that the total momentum before the collision is the same as the total momentum after the collision, and is expressed by the equation

$$m _ {1} v _ {0} = m _ {1} u _ {1} + m _ {2} u _ {2},$$

$u_{1}$ and $u_{2}$ the velocities of goblins fists and dwarfs head after collision.

Likewise, the conservation of the total kinetic energy is expressed by the equation

$$\frac {m _ {1} v _ {0} ^ {2}}{2} = \frac {m _ {1} u _ {1} ^ {2}}{2} + \frac {m _ {2} u _ {2} ^ {2}}{2}.$$

These equations may be solved directly to find $u_{1}$ and $u_{2}$

$$\left\{ \begin{array}{l} m _ {1} v _ {0} = m _ {1} u _ {1} + m _ {2} u _ {2} \\ m _ {1} v _ {0} ^ {2} = m _ {1} u _ {1} ^ {2} + m _ {2} u _ {2} ^ {2} \end{array} \right.$$ $$\left\{ \begin{array}{l} m _ {1} (v _ {0} - u _ {1}) = m _ {2} u _ {2} \\ m _ {1} (v _ {0} ^ {2} - u _ {1} ^ {2}) = m _ {2} u _ {2} ^ {2} \end{array} \right.$$ $$\left\{ \begin{array}{l} m _ {1} (v _ {0} - u _ {1}) = m _ {2} u _ {2} \\ m _ {1} (v _ {0} - u _ {1}) (v _ {0} + u _ {1}) = m _ {2} u _ {2} ^ {2} \end{array} \right.$$

Divide equations term by term

$$v _ {0} + u _ {1} = u _ {2}$$

Together with the first equation $m_{1}v_{0} = m_{1}u_{1} + m_{2}u_{2}$

$$u _ {1} = u _ {2} - v _ {0}$$ $$m _ {1} v _ {0} = m _ {1} \left(u _ {2} - v _ {0}\right) + m _ {2} u _ {2}$$ $$2 m _ {1} v _ {0} = u _ {2} \left(m _ {1} + m _ {2}\right)$$ $$u _ {2} = \frac {2 m _ {1} v _ {0}}{m _ {1} + m _ {2}} = \frac {2 \cdot 2 \cdot 5}{7} = 2.86 \mathrm{~m/s}.$$

Kinetic energy imparted to the dwarfs face by the one goblins fist

$$E _ {K 2} = \frac {m _ {2} u _ {2} ^ {2}}{2} = \frac {5 \cdot 20 ^ {2}}{2 \cdot 7 ^ {2}} = 20.41 \mathrm{~J}.$$

Answer. $E_{K2} = \frac{m_2u_2^2}{2} = \frac{5\cdot 20^2}{2\cdot 7^2} = 20.41\mathrm{~J}$.