Question

A 1000 kg rocket ship, travelling at \(100 \frac{\mathrm{m}}{\mathrm{s}},\) is acted upon by an average force of \(20 \mathrm{kN}\) applied in the direction of its motion for \(8 \mathrm{s}\) What is the change in velocity of the rocket? A. \(160 \frac{\mathrm{m}}{\mathrm{s}}\) B. \(160 \frac{\mathrm{m}}{\mathrm{s}}\) C. \(160,000 \frac{\mathrm{m}}{\mathrm{s}}\) \(160,000 \frac{\mathrm{m}}{\mathrm{s}}\)

Short answer

The change in velocity of the rocket is 160 m/s.

Step-by-step solution

- Identify the given quantities

The mass of the rocket ship is 1000 kg, the initial velocity is 100 m/s, the force applied is 20 kN, and the duration for which the force is applied is 8 s. Note that 1 kN is equivalent to 1000 N, so 20 kN is 20000 N.

- Use Newton's Second Law of Motion

Newton's Second Law states that the force applied to an object equals the mass of the object multiplied by its acceleration: \[ F = m \times a \]Let's rearrange this to solve for acceleration (\[ a = \frac{F}{m} \]).

- Calculate the acceleration

Substitute the force and mass into the equation: \[ a = \frac{20000 \; N}{1000 \; kg} = 20 \; \frac{m}{s^2} \]

- Determine the change in velocity

The change in velocity can be found using the equation: \[ \Delta v = a \times t \]Substitute the calculated acceleration and the given time into the equation: \[ \Delta v = 20 \; \frac{m}{s^2} \times 8 \; s = 160 \; \frac{m}{s} \]

- Verify the result

The change in velocity of the rocket is 160 m/s, which matches option A.

Key concepts

Newton's Second Law

To understand the motion of the rocket, we start with Newton's Second Law of Motion. This law states that the force applied to an object equals the mass of the object multiplied by its acceleration.
The equation is represented as:

\(F = m \times a \)
Here:

• \( F \) is the force applied
• \( m \) is the mass of the object
• \( a \) is the acceleration of the object

In the rocket problem, we have a force of \( 20 kN \) (which is \( 20000 N \)) applied to a rocket of mass \( 1000 kg \). To find the acceleration, we rearrange the equation to \( a = \frac{F}{m} \). Substituting in the values gives us the rocket's acceleration.

Acceleration Calculation

Let's calculate the acceleration of the rocket using the previously rearranged formula:
\( a = \frac{F}{m} \) Inserting the given values:
\( a = \frac{20000 \; N}{1000 \; kg} = 20 \; \frac{m}{s^2} \) 20 \( \frac{m}{s^2} \) is the acceleration of the rocket. This means that every second, the rocket's speed increases by \( 20 \; \frac{m}{s} \). This is important because the acceleration tells us how quickly the velocity of the rocket is changing over time due to the applied force.

Change in Velocity

Now, to determine the change in velocity (\( \) \( \Delta v \) ) of the rocket, we use the equation:
\( \Delta v = a \times t \) We already have the acceleration \( a \)= \( 20 \; \frac{m}{s^2} \) and the time \( t \) = 8 seconds. Substitute these values into the equation:
\( \Delta v = 20 \; \frac{m}{s^2} \times 8 \; s = 160 \; \frac{m}{s} \) This tells us that the velocity of the rocket changes by \( 160 \; \frac{m}{s} \) during the \( 8 \) seconds the force is applied. The initial velocity was not directly needed to determine how much the speed changes, but it is crucial in other contexts, such as finding the final velocity. The solution confirms that option A ( 160 \; \frac{m}{s} ) is correct.