Question

(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s?

(b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.

Short answer

(a) To have the same kinetic energy as a sprinter the elephant must move with\(1.47{\rm{ m}}/{\rm{s}}\).

(b) Metabolic rates of large animals are higher than humans as they require larger energy to move.

Step-by-step solution

Definition of Concept

Metabolic rate: The rate at which the body consumes energy to sustain life process is known as metabolic rate.

Kinetic energy: The energy stored in the body by the virtue of its motion is called kinetic energy. When a body is in motion, the energy associated with is in form of kinetic energy. According to the work-energy theorem, the work done on the body results in a change in the kinetic energy of the body.

Let us consider a body of mass\(m\)moving with velocity\(v\), then the kinetic energy associated with the body is,

\(KE = \frac{1}{2}m{v^2}\)

Find the velocity of the elephant

(a)

The kinetic energy of the elephant is,

\(K{E_e} = \frac{1}{2}{m_e}v_e^2\)

Here,\({m_e}\)is the mass of the elephant\(\left( {{m_e} = 3000{\rm{ kg}}} \right)\), and\({v_e}\)is the velocity of the elephant.

The kinetic energy of the sprinter is,

\(K{E_s} = \frac{1}{2}{m_s}v_s^2\)

Here,\({m_s}\)is the mass of the sprinter\(\left( {{m_s} = 65.0{\rm{ kg}}} \right)\), and\({v_s}\)is the velocity of the sprinter\(\left( {{v_s} = 10{\rm{ m}}/{\rm{s}}} \right)\).

Since, the kinetic energy of the elephant equals the kinetic energy of the sprinter. As a result,

\(\begin{aligned}{}K{E_e} &= K{E_s}\\\frac{1}{2}{m_e}v_e^2 &= \frac{1}{2}{m_s}v_s^2\end{aligned}\)

The expression for the velocity of the elephant is,

\({v_e} = \sqrt {\frac{{{m_s}}}{{{m_e}}}v_s^2} \)

Putting all known values,

\(\begin{aligned}{v_e} &= \sqrt {\frac{{\left( {65.0\;{\rm{kg}}} \right)}}{{\left( {3000{\rm{ kg}}} \right)}} \times {{\left( {10{\rm{ m}}/{\rm{s}}} \right)}^2}} \\ &= 1.47{\rm{ m}}/{\rm{s}}\end{aligned}\)

Therefore, the velocity of the elephant is \(1.47{\rm{ m}}/{\rm{s}}\).

Find the metabolic rate

(b)

Since, larger animals require very high energy to sustain life process.

Therefore, the required metabolic rates are higher than humans.