Question

Two parts of a spacecraft are separated by detonating the explosive bolts that hold them together. The masses of the parts are \(1200 \mathrm{~kg}\) and \(1800 \mathrm{~kg} ;\) the magnitude of the impulse delivered to each part is \(300 \mathrm{~N} \cdot \mathrm{s}\). What is the relative speed of separation of the two parts?

Short answer

The relative speed of separation of the two parts is \(0.417 ms^{-1}\).

Step-by-step solution

Calculate momentum for each part

The momentum \(p\) of an object is the product of its mass \(m\) and its velocity \(v\). This can be written as the following equation: \(p = m * v\). However, in this context, we are given the impulse \(J\), not the momentum. Impulse is the change in momentum, but as the initial velocities are 0, impulse equals momentum here. Therefore we find the momentum of each part by equating the impulse with momentum. Giving: \(p1 = J = 300 Ns\) for the \(1200kg\) part and \(p2 = J = 300 Ns\) for the \(1800kg\) part.

Calculate velocity for each part

We can find the velocities \(v1\) and \(v2\) of the two parts by rearranging the formula for momentum to solve for individual velocities. This leads to \(v = p/m\), thus \(v1 = p1/m1 = 300 Ns/1200kg = 0.25 ms^{-1}\) and \(v2 = p2/m2 = 300Ns / 1800kg = 0.167 ms^{-1}\).

Calculate relative speed of separation

The relative speed of separation of the two parts is the magnitude of the difference of the velocities. As the parts are moving in opposite directions, this would be \(v1 + v2\), which equals \(0.25 ms^{-1} + 0.167 ms^{-1} = 0.417 ms^{-1}\).

Key concepts

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion possessed by an object. It is calculated as the product of the mass of the object and its velocity, usually represented by the equation, \( p = m \times v \). Think of momentum as a measure of how difficult it would be to stop a moving object. A larger mass moving at a high speed has more momentum, meaning it requires a greater force or a longer time to bring it to rest.

An important aspect of understanding momentum is realizing that it is a vector quantity. This means it has both magnitude and direction. When we discuss the collision or separation of two objects, the direction of their momenta matters just as much as the amount. In the given exercise, each part of the spacecraft receives the same magnitude of impulse, but because the masses are different—their resulting momenta will also differ. The implication of this can be seen clearly when we try to calculate the relative speed of separation.

Impulse

Impulse in physics is often described as the change in momentum of an object. You can think of it as the effect of a force applied over a certain period of time. The impulse can be visualized as the area under a force-time graph and is mathematically defined as \( J = F \times \triangle t \), where \( F \) is the force applied, and \( \triangle t \) is the time duration over which the force is applied.

In the situation of the spacecraft parts separating, the impulse is the driving force behind the spacecraft's changing momentum. Given that the initial momenta were zero (as the spacecraft was intact and not moving in space), the impulse directly translates to the final momentum of each part. Impulse is extremely useful in collision problems as it helps us calculate the outcomes without needing to know the detailed forces during the interaction, as long as the total impulse is known.

Relative Speed

Relative speed is a concept that helps describe how fast one object is moving with respect to another. It's the rate at which the distance between two objects is changing. In the context of the spacecraft parts, they're moving away from each other, and we want to know the speed of separation.

When objects move in the same direction, the relative speed is the difference of their individual speeds. Conversely, when they move in opposite directions, their speeds are effectively added together to determine how quickly they're separating. An intuitive way to understand this is to imagine you're on a train moving at a certain speed, and another train passes you going the opposite direction. The speed at which the other train seems to zoom by would be the sum of both trains' speeds.

Conservation of Momentum

The conservation of momentum is a principle stating that the total momentum of a closed system remains constant provided no external forces are acting on it. This principle is rooted in Newton's third law, which tells us that for every action, there is an equal and opposite reaction.

In the case of the spacecraft scenario, the system consists of the two parts of the spacecraft. Before the detonation, the total system momentum is zero since they are stationary with respect to one another. As they separate, the total momentum must remain zero. To maintain this zero momentum, the momentum imparted to one part of the spacecraft must be equal and opposite to the momentum imparted to the other part. This is also evident in the calculated velocities of the individual parts and their subsequent relative speed of separation. Remember, the conservation of momentum is crucial for predicting the outcome of interactions in isolated systems.