Kinetic Energy — Question 9

Question 9

An energy budget constrains simultaneous speeds

Use classical translational kinetic energy $K=\tfrac12 m|\vec v|^2$. Two independent carts A and B have masses 1 kg and 2 kg. Their total kinetic energy is exactly 36 J in the same frame. Each must have speed at least 2 m/s.

1. Find the greatest possible speed of A.

2. Find the greatest possible speed of B.

3. Can both maxima occur simultaneously? Give the feasible relation between their speeds.

Question 9 - Solution

An energy budget constrains simultaneous speeds

(a) Reserve B’s minimum energy.

The governing relation is $$\boxed{\tfrac12m_Av_A^2+\tfrac12m_Bv_B^2=36\ \mathrm J}$$ B requires at least $$\tfrac12(2)2^2=4\ \mathrm J.$$ At most 32 J remains for A, so $$v_{A,\max}=\sqrt{2(32)/1}=8\ \mathrm{m/s},$$ attained when $$v_B=2\ \mathrm{m/s}.$$

(b) Reserve A’s minimum energy.

A requires 2 J; B can then have 34 J. Thus $$v_{B,\max}=\sqrt{34}\ \mathrm{m/s},$$ attained when $$v_A=2\ \mathrm{m/s}.$$

(c) Impose one joint constraint.

The feasible speeds obey $$\tfrac12v_A^2+v_B^2=36,$$ with both at least 2. The two separate maxima would require $$32+34=66\ \mathrm J,$$ so they cannot be attained together.