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Chapter 4: Problem 513

A neutron having mass of \(1.67 \times 10^{-27} \mathrm{~kg}\) and moving at \(10^{8} \mathrm{~m} / \mathrm{s}\) collides with a deuteron at rest and sticks to it. If the mass of the deuteron is \(3.34 \times 10^{-27} \mathrm{~kg}\) then the speed of the combination is (A) \(3.33 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (B) \(3 \times 10^{5} \mathrm{~m} / \mathrm{s}\) (C) \(33.3 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (D) \(2.98 \times 10^{5} \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
The short answer is (A) \(3.33 \times 10^{7} \mathrm{~m} / \mathrm{s}\).

Step by step solution

01

Determine the initial linear momentums of the neutron and deuteron

To find the initial linear momentum of the neutron, we'll use the formula: Neutron's Initial Linear Momentum = Neutron's mass × Neutron's initial velocity For the deuteron, since it is at rest, its initial linear momentum is zero.
02

Calculate the total initial linear momentum

We need to add the neutron's initial linear momentum to the deuteron's initial linear momentum (which is zero) to find the total initial linear momentum: Total Linear Momentum (Initial) = Neutron's Initial Linear Momentum
03

Determine the final linear momentum of the combined objects

After the collision, the neutron and deuteron stick together and move as one object. We can express their final linear momentum as: Final Linear Momentum = (Neutron's mass + Deuteron's mass) × Final speed
04

Apply the law of conservation of linear momentum

According to the conservation of linear momentum, the total initial linear momentum equals the final linear momentum. Set the expressions from Step 2 and Step 3 equal to each other, then solve for the final speed: Neutron's Initial Linear Momentum = (Neutron's mass + Deuteron's mass) × Final speed
05

Calculate the final speed and find the correct answer

Plug in the known values for the neutron's mass, neutron's initial velocity, and the deuteron's mass into the equation from Step 4. Then, solve for the final speed and find the matching option: \(1.67 \times 10^{-27} kg \times 10^8 m/s = (1.67 \times 10^{-27} kg + 3.34 \times 10^{-27} kg) × Final \: speed\) The final speed can be calculated as: Final speed = \(\frac{1.67 \times 10^{-27} kg \times 10^8 m/s}{(1.67 \times 10^{-27} kg + 3.34 \times 10^{-27} kg)}\) By calculating this equation, we get the final speed and we can find the correct answer among the given options: Final speed ≈ 3 × 10^7 m/s The correct answer is (A) \(3.33 \times 10^{7} \mathrm{~m} / \mathrm{s}\).

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