COMEDK 2023 Shift 1 Paper

To find the time the helium nucleus is in the tube, we can use the kinematic equations that describe motion under uniform acceleration. The relevant equation in this setting, given the initial velocity (vi), final velocity (vf), displacement (s), and acceleration (a), is:
vf=vi+at
However, we don't directly know the acceleration or the time, so we'll use another equation that relates these variables without the need for acceleration. This equation is:
s=vit+‌

1

2

at2
Given that we know the initial and final velocities and the displacement, but still lack the acceleration, we can use a different form that makes use of both initial and final velocities:
s=‌

vi+vf

2

⋅t
This equation comes from averaging the initial and final velocities in a uniformly accelerated motion to find the average velocity, then multiplying by the time to find the displacement.
We know from the problem that:
The length of the tube, which is the displacement s=4m,
The initial velocity vi=2000m∕ s,
And the final velocity vf=8000m∕ s.
Plugging these values into our formula, we have:
4=‌

2000+8000

2

⋅t
This simplifies to:
4=‌

10000

2

⋅t
Which further simplifies to:
4=5000t
Solving for t gives:
t=‌

4

5000

=0.0008 s
Hence, the time the particle is in the tube is 0.0008 seconds or 8×10−4 s, which corresponds to Option C.