I saw this problem half answered already but the answers I got were wrong
A 238U nucleus is moving in the x-direction at 4.8×105m/s when it decays into an alpha particle (4He) and a 234Th nucleus. The alpha moves at 1.2×107m/s at 36∘ above the x-axis.
a. Find the speed of the thorium nucleus. v=
b. Find the direction of the motion of the thorium nucleus.
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To solve this problem, we will use the principles of conservation of momentum and conservation of kinetic energy. Let's break it down step by step.
a. Finding the speed of the thorium nucleus (v):
To solve for the speed of the thorium nucleus, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the decay should be equal to the total momentum after the decay.
The total momentum before the decay (p_initial) is the momentum of the uranium nucleus (238U), and it can be calculated using the formula:
p_initial = mass * velocity
Given that the mass of the 238U nucleus is 238 atomic mass units (u) and the velocity is 4.8 × 10^5 m/s, we can calculate p_initial.
Next, applying the principle of conservation of momentum, the total momentum after the decay (p_final) should be equal to the sum of the momentum of the alpha particle (4He) and the thorium nucleus (234Th).
p_final = p_alpha + p_thorium
Given that the mass and velocity of the alpha particle (4He) are known, we can calculate its momentum (p_alpha).
Finally, we can use the conservation of momentum equation to find the momentum of the thorium nucleus (p_thorium).
Now, since momentum (p) is defined as mass (m) multiplied by velocity (v), we can rearrange the equation:
p = m * v
Thus, the speed of the thorium nucleus (v) can be found by dividing its momentum (p_thorium) by its mass (m_thorium).
b. Finding the direction of motion of the thorium nucleus:
To find the direction of motion of the thorium nucleus, we consider the direction of the alpha particle's motion after the decay.
According to the given information, the alpha particle moves at 1.2 × 10^7 m/s at 36° above the x-axis. So, the direction of motion of the thorium nucleus will be in the same direction as the alpha particle but with a 180° shift (opposite direction).
I hope this explanation helps!