Question

A rocket is fired from the earth surface with constant vertical acceleration. A bolt is dropped from the rocket 2.0 s after the firing and fuel of the rocket is finished in 7.0 s after the bolt is dropped. Time of free fall of the bolt to the ground is 4.8 s. Acceleration of the free fall is 10 m/s². Which of the following statement is/are correct?

• A. Ascending acceleration of the rocket with fuel is nearly 10.0 m/s²
• B. Maximum height attained by the bolt from the ground is nearly 40 m
• C. Maximum speed of the rocket is nearly 70 m/s
• D. Both (1) and (2)

Answer 1

Answer: (A), (B)

Both (1) and (2)

Answer 2

Detailed Solution:

Let a be the constant upward acceleration of the rocket.
Let g be the acceleration due to gravity, given as 10 m/s².

1. Analyze the motion of the bolt:
The bolt is dropped 2.0 s after the rocket is fired.
At t = 2.0 s:

• The velocity of the rocket (and thus the bolt at the moment of dropping) is u_bolt = a * t = a * 2.0 = 2a. This velocity is directed upwards.
• The height of the rocket (and thus the bolt at the moment of dropping) from the ground is h_drop = 0.5 * a * t^2 = 0.5 * a * (2.0)^2 = 2a.

After being dropped, the bolt undergoes free fall under gravity. Its initial velocity is u_bolt = 2a upwards, and the acceleration is g = -10 m/s² (taking upward as positive).
The time of free fall of the bolt to the ground is given as T_fall = 4.8 s.
The displacement of the bolt during its free fall is -h_drop (since it starts at h_drop and ends at the ground, which is 0).

Using the equation of motion: s = ut + 0.5gt^2
-h_drop = u_bolt * T_fall + 0.5 * (-g) * T_fall^2
Substitute the values:
-2a = (2a) * (4.8) - 0.5 * (10) * (4.8)^2
-2a = 9.6a - 5 * (23.04)
-2a = 9.6a - 115.2
115.2 = 9.6a + 2a
115.2 = 11.6a
a = 115.2 / 11.6
a ≈ 9.931 m/s²

2. Evaluate Statement (1): Ascending acceleration of the rocket with fuel is nearly 10.0 m/s²
From our calculation, a ≈ 9.931 m/s². This value is very close to 10.0 m/s².
Therefore, statement (1) is correct.

3. Evaluate Statement (2): Maximum height attained by the bolt from the ground is nearly 40 m
The bolt is dropped from a height h_drop = 2a = 2 * (115.2 / 11.6) ≈ 19.862 m.
Its initial upward velocity when dropped is u_bolt = 2a ≈ 19.862 m/s.
The bolt will continue to move upwards against gravity until its velocity becomes zero, and then it will fall.
The additional height gained by the bolt after being dropped (before it starts falling downwards) can be calculated using v^2 = u^2 + 2gs:
0 = u_bolt^2 + 2 * (-g) * h_gain
h_gain = u_bolt^2 / (2g) = (19.862)^2 / (2 * 10)
h_gain = 394.49 / 20 ≈ 19.724 m
The maximum height attained by the bolt from the ground is H_max_bolt = h_drop + h_gain.
H_max_bolt = 19.862 + 19.724 = 39.586 m.
This value is nearly 40 m.
Therefore, statement (2) is correct.

4. Evaluate Statement (3): Maximum speed of the rocket is nearly 70 m/s
The fuel of the rocket finishes 7.0 s after the bolt is dropped.
Time of bolt drop = 2.0 s from firing.
So, the fuel finishes at t_burnout = 2.0 s + 7.0 s = 9.0 s from firing.
The rocket accelerates at a constant rate a until the fuel runs out. Thus, its maximum speed occurs at the moment the fuel finishes.
Maximum speed of the rocket v_max_rocket = a * t_burnout = (115.2 / 11.6) * 9.0
v_max_rocket ≈ 9.931 * 9.0 ≈ 89.379 m/s.
This value is not nearly 70 m/s.
Therefore, statement (3) is incorrect.

Conclusion:
Statements (1) and (2) are correct, while statement (3) is incorrect.
Option (4) states "Both (1) and (2)". Since both (1) and (2) are correct, option (4) is the correct choice.

The last line "4. If v is speed, r is radius and g is acceleration due to gravity, which of the following is dimensionless? (1) v^2r/g" appears to be an unrelated question and should be disregarded.

The final answer is $\boxed{\text{Both (1) and (2)}}$

Explanation of the solution (minimal):

1. Calculate rocket acceleration 'a' using the bolt's motion: Bolt dropped at t=2s with u=2a and h=2a. It falls for 4.8s. Using s = ut + 0.5gt^2, -2a = (2a)(4.8) - 0.5(10)(4.8)^2, which gives a ≈ 9.93 m/s². So (1) is correct.
2. Calculate bolt's maximum height: Bolt's initial upward velocity u_bolt = 2a ≈ 19.86 m/s. Height at drop h_drop = 2a ≈ 19.86 m. Additional height gained h_gain = u_bolt^2 / (2g) ≈ (19.86)^2 / 20 ≈ 19.72 m. Total max height H_max = h_drop + h_gain ≈ 19.86 + 19.72 = 39.58 m ≈ 40 m. So (2) is correct.
3. Calculate rocket's maximum speed: Fuel finishes at t = 2s + 7s = 9s. Max speed v_max = a * 9s ≈ 9.93 * 9 = 89.37 m/s. So (3) is incorrect.
4. Since (1) and (2) are correct, option (4) is the answer.