Question

Two astronauts are playing catch in a zero gravitational field. Astronaut 1 of mass m1 is initially moving to the right with speed v1 . Astronaut 2 of mass m2 is initially moving to the right with speed v2>v1 . Astronaut 1 throws a ball of mass m with speed u relative to herself in a direction opposite to her motion. Astronaut 2 catches the ball. The final speed of astronaut 1 is vf,1 and the final speed of astronaut 2 is vf,2 .What is the speed vf,1 of astronaut 1 after throwing the ball?

Answer 1

The final velocity ($v_1_f$) of the first astronaut will be greater than the final velocity of the second astronaut ($v_2_f$) to ensure that the total initial momentum of both astronauts is equal to the total final momentum of both astronauts after throwing the ball.

The given parameters;

• Mass of the first astronaut, = m₁
• Mass of the second astronaut, = m₂
• Initial velocity of the first astronaut, = v₁
• Initial velocity of the second astronaut, = v₂ > v₁
• Mass of the ball, = m
• Speed of the ball, = u
• Final velocity of the first astronaut, = $v_f_1$
• Final velocity of the second astronaut, = $v_f_2$

The final velocity of the first astronaut relative to the second astronaut after throwing the ball is determined by applying the principle of conservation of linear momentum.

$m_1v_1 + m_2v_2 = m_2v_2_f + m_1v_1_f$

if v₂ > v₁, then $v_1_f > v_2_f$, to conserve the linear momentum.

Thus, the final velocity ($v_1_f$) of the first astronaut will be greater than the final velocity of the second astronaut ($v_2_f$) to ensure that the total initial momentum of both astronauts is equal to the total final momentum of both astronauts after throwing the ball.

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