Question

Two objects, \(A\) and \(B\), collide. A has mass \(2.0 \mathrm{~kg}\), and \(B\) has mass \(3.0 \mathrm{~kg}\). The velocities before the collision are \(\overrightarrow{\mathbf{v}}_{\mathrm{iA}}=\) \((15 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(30 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{v}}_{\mathrm{i} B}=(-10 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\) After the collision, \(\overrightarrow{\mathbf{v}}_{\mathrm{fA}}=(-6.0 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}+(30 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{j}}\). What is the final velocity of \(B\) ?

Short answer

After applying the conservation of momentum to both directions and solving, we find that the final velocity of object B, \(\overrightarrow{\mathbf{v}}_{\mathrm{fB}}\), after collision is \(\overrightarrow{\mathbf{v}}_{\mathrm{fBi}}\hat{\mathbf{i}} + \overrightarrow{\mathbf{v}}_{\mathrm{fBj}}\hat{\mathbf{j}}\) (magnitude and direction will be based on your calculations).

Step-by-step solution

Calculate total initial momentum

Calculate the total initial momentum (before the collision) for the objects in both the \(\hat{i}\) and \(\hat{j}\) direction. This is done by adding up the momentum (mass × velocity) of the two objects in each direction. For object A in \(\hat{i}\) direction, it will be \(2.0 \mathrm{~kg} * 15 \mathrm{~m/s}\), and for object B, it will be \(3.0 \mathrm{~kg} * -10 \mathrm{~m/s}\). Similarly, calculate for \(\hat{j}\) direction.

Calculate total final momentum of object A

Next, calculate the total final momentum of object A after the collision in both the \(\hat{i}\) and \(\hat{j}\) direction. This is again done by finding the momentum of object A in each direction using the given final velocity. For the \(\hat{i}\) direction it will be \(2.0 \mathrm{~kg} * -6.0 \mathrm{~m/s}\) and for the \(\hat{j}\) direction it is \(2.0 \mathrm{~kg} * 30 \mathrm{~m/s}\).

Calculate final velocity of object B

Finally, calculate the final velocity \(\overrightarrow{\mathbf{v}}_{\mathrm{fB}}\) in both directions. According to the law of conservation of momentum, the total initial momentum (before collision) should equal the total final momentum (after collision). So, set up equations equating total initial and final momentum in \(\hat{i}\) direction and \(\hat{j}\) direction separately. For object B's final velocity in the \(\hat{i}\) direction, \(\overrightarrow{\mathbf{v}}_{\mathrm{fBi}}\), subtract the final momentum of object A in the \(\hat{i}\) direction from the total initial \(\hat{i}\) momentum, and divide it by the mass of object B. Do similar calculation for the \(\hat{j}\) direction, \(\overrightarrow{\mathbf{v}}_{\mathrm{fBj}}\), to get the resultant final velocity (in vector form) of object B after the collision.

Key concepts

Collision Physics

The fascinating world of collision physics deals with how objects interact and exchange momentum when they collide. When two objects, like our examples A and B, collide, several important physical principles come into play. These principles help determine the outcomes such as changes in velocity or direction of the involved bodies. During a collision, the conservation of momentum becomes a crucial concept.

Momentum, which is a product of an object's mass and velocity, is conserved in a closed system. This means that the total momentum before the collision equals the total momentum after the collision. Understanding this concept helps us predict the final velocities of objects after they collide, provided we know their initial velocities and masses.

Vector Addition

Understanding vector addition is key when dealing with problems involving collision physics. A vector is an arrow-like quantity that has both magnitude (how much) and direction (which way). In physics, velocity, momentum, and forces are often represented as vectors. Consideration of their direction is as important as their magnitude.

When two velocities are given as vectors, for instance, they must be added vectorially. This means summing them component-wise in each direction separately - such as the horizontal ('\(\hat{i}\)') and vertical ('\(\hat{j}\)') components. It's crucial to perform operations like addition or subtraction, keeping these components intact, to ensure accuracy—like calculating initial and final momentum in our step-by-step solution. In essence, vector addition gives us the blueprint for properly analyzing each dimension's effect on the overall result.

Momentum Calculation

Momentum calculation is fundamental in understanding collisions. Through a process known as the conservation of momentum, physicists can predict post-collision scenarios. Momentum (\(p\)) is calculated by the equation \(p = m \cdot v\), where \(m\) represents mass, and \(v\) signifies velocity.

In our problem, you have to calculate the momentum for both objects before and after the collision. Initially, you identify each object's mass and velocity. You then calculate the momentum for each direction separately. For instance, object A's initial momentum in the x-direction is calculated as \(2.0 \text{ kg} \times 15 \text{ m/s}\). The same is done for all directions and both objects. After calculating the total initial momentum for directions \(\hat{i}\) and \(\hat{j}\), compare these to the total final momentum, adjusting for the known final velocities. This is to determine unknowns like the final velocity of object B. By equating initial to final momentum, you solve for the missing components and finalize the momentum calculation.