The best way to analyze this situation is to apply the principle of conservation of momentum and understand how momentum changes for individual bodies in a collision. Momentum, defined as the product of mass and velocity ([katex] p = mv [/katex]), is always conserved in an isolated system (without external forces).

Given:
– Let [katex] v_t [/katex] and [katex] v_c [/katex] be the velocities of the truck and the car respectively.
– Assume the truck has a mass [katex] m_t [/katex] and the car has a mass [katex] m_c [/katex].
– Truck’s initial velocity [katex] v_{t, \text{init}} = 15 , \text{km/h} [/katex] (we’ll consider the direction towards the car as positive).
– Car’s initial velocity [katex] v_{c, \text{init}} = -30 , \text{km/h} [/katex] (since it’s head-on and opposite, it’s negative).

Step 1: Calculate Initial Momenta
– Momentum of the truck initially: [katex] p_{t, \text{init}} = m_t \times 15 [/katex]
– Momentum of the car initially: [katex] p_{c, \text{init}} = m_c \times (-30) [/katex]

Step 2: Use Conservation of Momentum
– Total initial momentum [katex] p_{\text{total, init}} = m_t \times 15 + m_c \times (-30) [/katex].

The collision is an isolated event with no external forces, so total momentum must be preserved:
– [katex] p_{\text{total, final}} = p_{\text{total, init}} [/katex].

After the collision, depending on the details (elastic or inelastic), both truck and car will have new velocities [katex] v_{t, \text{final}} [/katex] and [katex] v_{c, \text{final}} [/katex] but their combined momentum remains [katex] m_t \times v_{t, \text{final}} + m_c \times v_{c, \text{final}} = m_t \times 15 + m_c \times (-30) [/katex].

Step 3: Change in Momentum
– Change in momentum for the truck: [katex] \Delta p_t = m_t \times (v_{t, \text{final}} – 15) [/katex]
– Change in momentum for the car: [katex] \Delta p_c = m_c \times (v_{c, \text{final}} + 30) [/katex]

Change in the magnitude of the individual momenta will depend on the masses of the truck and car and their change in velocities but must reflect conservation principles.

Answer Analysis:
– (a) Incorrect – A greater mass does not directly mean a greater change in momentum unless the change in velocity is considered.
– (b) Correct – Since momentum is conserved and the system is isolated, both the car and truck experience the equal magnitude of momentum change but in opposite directions.
– (c) Incorrect – Greater initial speed of the car implies greater initial momentum, but not necessarily a greater change in momentum.
– (d) Incorrect – Both vehicles change their momentum, but the total system momentum is conserved.
– (e) Incorrect – Statement (b) is necessarily true considering the law of conservation of momentum in an isolated system.

Thus, the statement that best describes the situation is (b) They both have the same change in magnitude of momentum because momentum is conserved.