Question

two identical train cars are on the same track. the first train car is moving with a speed of 10 m/s toward the stationary second train car. if they get hooked together upon contact, how fast will the newly connected trains be moving? view available hint(s) hint 1: consider conservation of momentum during collisions. the connected trains will be moving 5 m/s. the speed of the connected trains will stay constant. the connected trains will be moving faster. the connected trains will be moving slower.

Explanation:

Step1: Recall Conservation of Momentum

The law of conservation of momentum states that for a closed system, the total initial momentum equals the total final momentum. The formula is $m_1v_1 + m_2v_2=(m_1 + m_2)v_f$, where $m_1,m_2$ are masses, $v_1,v_2$ are initial velocities, and $v_f$ is the final velocity.

Step2: Define Variables

Let the mass of each train car be $m$ (since they are identical). The first train's initial velocity $v_1 = 10\,\text{m/s}$, the second train's initial velocity $v_2 = 0\,\text{m/s}$ (stationary).

Step3: Apply Conservation of Momentum

Substitute into the formula: $m\times10 + m\times0=(m + m)v_f$. Simplify left side: $10m = 2m\times v_f$.

Step4: Solve for $v_f$

Divide both sides by $2m$: $v_f=\frac{10m}{2m}=5\,\text{m/s}$. Also, since the mass increases (two cars now), the velocity should be less than the initial velocity of the first car (10 m/s), so it moves slower than the original first car's speed.

Answer:

A. The connected trains will be moving 5 m/s.
D. The connected trains will be moving slower.

(Note: Both A and D are correct. A gives the numerical speed, D describes the comparison to the initial speed of the first car. If only one is to be chosen as the most probable for the "how fast" question, A is more direct about the speed value, but D is also correct about the qualitative change. However, based on the hint and calculation, A is the numerical answer from momentum conservation, and D is also correct as 5 m/s is slower than 10 m/s. If the question expects the speed value, A is the key numerical answer.)