A shell of total mass 5.0\text{ kg} is moving horizontally with a velocity of 20\text{ m s… — JAMB UTME

A shell of total mass $5.0\text{ kg}$ is moving horizontally with a velocity of $20\text{ m s}^{-1}$ when an internal explosion splits it into two fragments. One fragment of mass $2.0\text{ kg}$ is propelled backward along the original path at a speed of $10\text{ m s}^{-1}$ . What is the magnitude of the velocity, in $\text{m s}^{-1}$ , of the second fragment immediately after the explosion?

Answer: 40 m s^-1

Answer

The magnitude of the velocity of the second fragment immediately after the explosion is

40\text{ m s}^{-1}

According to the Law of Conservation of Linear Momentum, the total momentum of a system remains constant when no net external horizontal force acts on it. Taking the original direction of motion as positive, the initial momentum is

100\text{ kg m s}^{-1}

. Since the

2.0\text{ kg}

fragment moves backward at

10\text{ m s}^{-1}

, its momentum is

-20\text{ kg m s}^{-1}

. For total momentum to remain

+100\text{ kg m s}^{-1}

, the remaining

3.0\text{ kg}

fragment must carry a momentum of

+120\text{ kg m s}^{-1}

, which corresponds to a velocity of

40\text{ m s}^{-1}

Step-by-Step Solution

Calculate the initial momentum of the shell prior to the explosion.

p_i = 5.0\text{ kg} \times 20\text{ m s}^{-1} = 100\text{ kg m s}^{-1}

in the initial forward direction.

Before the internal explosion, the system consists of a single mass moving with a constant velocity.

Apply the Law of Conservation of Linear Momentum taking vector direction into account.

p_i = m_1 v_1 + m_2 v_2 \implies 100 = 2.0(-10) + 3.0 v_2

In the absence of external forces, total momentum is conserved. The fragment propelled backward takes a negative sign relative to the initial forward motion.

Solve the algebraic equation for the unknown velocity

v_2

100 + 20 = 3.0 v_2 \implies 120 = 3.0 v_2 \implies v_2 = 40\text{ m s}^{-1}

Isolating

v_2

gives the forward velocity magnitude of the remaining

3.0\text{ kg}

piece.

Key Concept

Conservation of Linear Momentum in Explosions (1D Vector Sign Convention)

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