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95ICHS (1) m,v, = (m, + m,) Apply the isolated system model for momentum in the y-direction: (2) mzv2 sin e Divide Equation (2) by Equation (1):

95ICHS (1) m,v, = (m, + m,) Apply the isolated system model for momentum in the y-direction: (2) mzv2 sin e Divide Equation (2) by Equation (1):

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**Collision at an Intersection** **Problem:** A 1,550 kg car traveling east with a speed of 28.1 m/s collides at an intersection with a 2,550 kg truck traveling north at a speed of 20.3 m/s as shown in the figure. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision. **Diagram Explanation:** The figure shows a top-down view of an intersection with a car moving to the east and a truck moving to the north. The point of collision is marked, and the motion of both vehicles is indicated with arrows. **SOLUTION** **Conceptualize:** The figure should help you conceptualize the situation before and after the collision. Let us choose east to be along the positive x-direction and north to be along the positive y-direction. - What direction is the blue car moving initially? - To the north - To the south - To the east - To the west - *Correct answer: To the east* **Categorize:** Because we consider moments immediately before and immediately after the collision as defining our time interval, we ignore the small effects that friction would have on the wheels of the vehicles and model the two vehicles as an isolated system in terms of momentum. We also ignore the vehicles' sizes and model them as particles. The collision is perfectly *inelastic* because the car and the truck stick together after the collision. **Analyze:** Before the collision, the only object having momentum in the x-direction is the *car*. Therefore, the magnitude of the total initial momentum of the system (car plus truck) in the x-direction is that of only the car. Similarly, the total initial momentum of the system in the y-direction is that of the *truck*. After the collision, let us assume the wreckage moves at an angle θ with respect to the x-axis with speed v_f. *(For the next three answers, use the following as necessary: `m1, m2, vx, and vy.` Do not substitute numerical values; use variables only.)* Apply the isolated system model for momentum in the x-direction: Δpx = 0 = Σpfx - Σpix ⇒ m1vx = (m1 + m2)vfx *(Equation 1)* Apply the isolated system model for momentum in the y-direction: Δpy = 0
Transcribed Image Text:**Collision at an Intersection** **Problem:** A 1,550 kg car traveling east with a speed of 28.1 m/s collides at an intersection with a 2,550 kg truck traveling north at a speed of 20.3 m/s as shown in the figure. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision. **Diagram Explanation:** The figure shows a top-down view of an intersection with a car moving to the east and a truck moving to the north. The point of collision is marked, and the motion of both vehicles is indicated with arrows. **SOLUTION** **Conceptualize:** The figure should help you conceptualize the situation before and after the collision. Let us choose east to be along the positive x-direction and north to be along the positive y-direction. - What direction is the blue car moving initially? - To the north - To the south - To the east - To the west - *Correct answer: To the east* **Categorize:** Because we consider moments immediately before and immediately after the collision as defining our time interval, we ignore the small effects that friction would have on the wheels of the vehicles and model the two vehicles as an isolated system in terms of momentum. We also ignore the vehicles' sizes and model them as particles. The collision is perfectly *inelastic* because the car and the truck stick together after the collision. **Analyze:** Before the collision, the only object having momentum in the x-direction is the *car*. Therefore, the magnitude of the total initial momentum of the system (car plus truck) in the x-direction is that of only the car. Similarly, the total initial momentum of the system in the y-direction is that of the *truck*. After the collision, let us assume the wreckage moves at an angle θ with respect to the x-axis with speed v_f. *(For the next three answers, use the following as necessary: `m1, m2, vx, and vy.` Do not substitute numerical values; use variables only.)* Apply the isolated system model for momentum in the x-direction: Δpx = 0 = Σpfx - Σpix ⇒ m1vx = (m1 + m2)vfx *(Equation 1)* Apply the isolated system model for momentum in the y-direction: Δpy = 0
**Transcription for Educational Website** --- **Solve for θ (in degrees counterclockwise from the +x-axis) and substitute numerical values:** \[ \theta = \tan^{-1} \left(\frac{m_2 v_{2i}}{m_1 v_{1i}}\right) \] ° counterclockwise from the +x-axis Use Equation (2) to find the value of \( v_f \) and substitute numerical values (Enter the magnitude in m/s): \[ v_f = \frac{m_2 v_{i}}{(m_1 + m_2) \sin \theta} \] / m/s --- **Finalize** Notice that the final speed of the combination is less than the initial speeds of the two cars. This result is consistent with the kinetic energy of the system being reduced in an inelastic collision. It might help if you draw the diagram of the vectors before the collision and the momentum angle after the collision. --- **EXERCISE** The eastbound car in the example is driven by an off-duty police officer, who knows her speed was exactly 20.0 m/s. From studying the angle θ (and the mass of the two vehicles), she concluded that the truck was over the speed limit of 80 km/h driving on that secondary road. Of course, the truck also failed to stop before entering the main road. If the truck were not speeding, what should be the maximum angle for θ (in degrees counterclockwise from the +x-axis) in this accident? ![Hint Button] --- *The answer box contains the submission of the angle as 99.91°, with an indication that this submission was incorrect (as shown by the red cross).* - **Comment:** Use your diagram and the relations between the components of velocity before and just after the collision to find the angle. ° counterclockwise from the +x-axis. ![Need Help? Button] --- *This problem is presented on a webpage from a physics course, specifically PHY2048 for Fall 2021 via WebAssign.*
Transcribed Image Text:**Transcription for Educational Website** --- **Solve for θ (in degrees counterclockwise from the +x-axis) and substitute numerical values:** \[ \theta = \tan^{-1} \left(\frac{m_2 v_{2i}}{m_1 v_{1i}}\right) \] ° counterclockwise from the +x-axis Use Equation (2) to find the value of \( v_f \) and substitute numerical values (Enter the magnitude in m/s): \[ v_f = \frac{m_2 v_{i}}{(m_1 + m_2) \sin \theta} \] / m/s --- **Finalize** Notice that the final speed of the combination is less than the initial speeds of the two cars. This result is consistent with the kinetic energy of the system being reduced in an inelastic collision. It might help if you draw the diagram of the vectors before the collision and the momentum angle after the collision. --- **EXERCISE** The eastbound car in the example is driven by an off-duty police officer, who knows her speed was exactly 20.0 m/s. From studying the angle θ (and the mass of the two vehicles), she concluded that the truck was over the speed limit of 80 km/h driving on that secondary road. Of course, the truck also failed to stop before entering the main road. If the truck were not speeding, what should be the maximum angle for θ (in degrees counterclockwise from the +x-axis) in this accident? ![Hint Button] --- *The answer box contains the submission of the angle as 99.91°, with an indication that this submission was incorrect (as shown by the red cross).* - **Comment:** Use your diagram and the relations between the components of velocity before and just after the collision to find the angle. ° counterclockwise from the +x-axis. ![Need Help? Button] --- *This problem is presented on a webpage from a physics course, specifically PHY2048 for Fall 2021 via WebAssign.*
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