Problem 56 - College Physics

Question

\(\bullet\) In outer space, where gravity is negligible, a \(75,000 \mathrm{kg}\) rocket (including \(50,000 \mathrm{kg}\) of fuel) expels this fuel at a steady rate of 135 \(\mathrm{kg} / \mathrm{s}\) with a speed of 1200 \(\mathrm{m} / \mathrm{s}\) relative to the rocket. (a) Find the thrust of the rocket. (b) What are the initial acceleration and the maximum acceleration of the rocket? (c) After the fuel runs out, what happens to this rocket's acceleration? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning. (d) After the fuel runs out, what happens to the rocket's speed? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning.

Step-by-Step Solution

Answer

(a) Thrust is 162,000 N. (b) Initial acceleration is 2.16 m/s²; max is 6.48 m/s². (c) Acceleration becomes zero. (d) Speed remains the same.

Step 1: Find the Thrust of the Rocket

The thrust of a rocket is given by the formula \( F = v \times \dot{m} \), where \( v \) is the speed of the expelled fuel relative to the rocket, and \( \dot{m} \) is the rate of fuel expulsion. Here, \( v = 1200 \, \text{m/s} \) and \( \dot{m} = 135 \, \text{kg/s} \). Substituting these values, we calculate the thrust: \( F = 1200 \, \text{m/s} \times 135 \, \text{kg/s} = 162,000 \, \text{N} \).

Step 2: Calculate Initial Acceleration

Using Newton's second law, \( F = ma \), the initial acceleration can be calculated by rearranging the formula to \( a = \frac{F}{m} \). The initial total mass \( m = 75,000 \, \text{kg} \). With the calculated thrust \( F = 162,000 \, \text{N} \), we have \( a = \frac{162,000 \, \text{N}}{75,000 \, \text{kg}} = 2.16 \, \text{m/s}^2 \).

Step 3: Calculate Maximum Acceleration

Maximum acceleration occurs just before the fuel runs out when the mass is minimum. Then the total mass is \( 25,000 \, \text{kg} \) (since the rocket has only its empty mass left). Substituting in the formula \( a = \frac{162,000 \, \text{N}}{25,000 \, \text{kg}} = 6.48 \, \text{m/s}^2 \).

Step 4: Analyze Rocket's Acceleration After Fuel is Expelled

Once the fuel is depleted, the thrust drops to zero since there is no fuel to expel. Therefore, the acceleration becomes zero suddenly. Thus, the answer is (ii) it suddenly becomes zero.

Step 5: Analyze Rocket's Speed After Fuel is Expelled

After the fuel runs out, there will be no more thrust. However, in the absence of opposing forces, the speed that the rocket had just before the fuel depletion is maintained due to inertia. Thus, the speed remains the same as it was when the fuel ran out, so the answer is (i) it remains the same as it was just as the fuel ran out.

Key Concepts

Thrust Calculation

Thrust is the force that propels a rocket forward. To calculate the thrust produced by a rocket engine, we can use the formula:

• Thrust, \( F = v \times \dot{m} \),

where \( v \) is the speed at which the fuel is expelled relative to the rocket, and \( \dot{m} \) is the rate at which fuel mass is expelled. In our scenario, the exhaust speed \( v \) is 1200 meters per second, and the fuel exhalation rate \( \dot{m} \) is 135 kilograms per second.
Substitute these values into the formula to find the thrust:

• \( F = 1200 \, \text{m/s} \times 135 \, \text{kg/s} = 162,000 \, \text{N} \) (Newtons).

This signifies that the rocket engine can produce a thrust of 162,000 Newtons, driving the rocket forward at a high velocity.

Newton's Second Law

Newton's second law of motion is crucial in determining how forces affect a rocket's motion. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration:

• \( F = ma \),

where \( F \) is the force or thrust, \( m \) is the mass of the rocket, and \( a \) is the acceleration. For our rocket:

• Initial mass of the rocket, \( m = 75,000 \, \text{kg} \), and,
• Thrust, \( F = 162,000 \, \text{N} \).

Using the formula, we rearrange to find the acceleration:

• \( a = \frac{F}{m} = \frac{162,000 \, \text{N}}{75,000 \, \text{kg}} = 2.16 \, \text{m/s}^2 \).

This tells us the initial acceleration of our rocket given the force it produces.

Acceleration Change

The acceleration of a rocket doesn’t stay constant; it varies as fuel burns and the mass of the rocket changes. The initial acceleration when the rocket is fully fueled is given by the formula:

• \( a_\text{initial} = \frac{162,000 \, \text{N}}{75,000 \, \text{kg}} = 2.16 \, \text{m/s}^2 \).
• Maximum acceleration is achieved when the rocket has less mass to propel, just before the fuel completely runs out. At this point, the rocket's mass is 25,000 kg, resulting in:
• \( a_\text{max} = \frac{162,000 \, \text{N}}{25,000 \, \text{kg}} = 6.48 \, \text{m/s}^2 \).

The acceleration of the rocket increases as the rocket's mass decreases. However, once the fuel depletes, thrust becomes zero, and hence, the acceleration drops suddenly to zero.

Inertia

Inertia is the property of an object to resist changes in its state of motion. After the fuel runs out, the rocket no longer has thrust to propel it forward. Nevertheless, due to inertia, it continues moving at the velocity it had the moment fuel depletion occurred. This implies:

• The rocket's speed remains constant once the fuel is exhausted, as there is no net external force acting to alter its motion.

In space, with negligible air resistance or gravity, the rocket will continue at the same speed indefinitely unless acted upon by another force. This continuity of motion is a direct consequence of the principle of inertia.