A 1000-kg car is travelling west at 17.0 m/s toward an intersection. A 1500-kg truck is travelling north at 20.0 m/s toward the same intersection. The car and truck collide at the intersection and sick together. What is the angle of their velocity after the collision relative to the east direction?
Question
A 1000-kg car is travelling west at 17.0 m/s toward an intersection. A 1500-kg truck is travelling north at 20.0 m/s toward the same intersection. The car and truck collide at the intersection and sick together. What is the angle of their velocity after the collision relative to the east direction?
Solution
To solve this problem, we need to use the law of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Step 1: Calculate the momentum of each vehicle before the collision. The momentum (p) of an object is calculated by multiplying its mass (m) by its velocity (v). So, for the car, p_car = m_car * v_car = 1000 kg * 17.0 m/s = 17000 kgm/s. For the truck, p_truck = m_truck * v_truck = 1500 kg * 20.0 m/s = 30000 kgm/s.
Step 2: Determine the direction of each vehicle's momentum. The car is traveling west, so its momentum is directed west. The truck is traveling north, so its momentum is directed north.
Step 3: Combine the momenta to find the total momentum before the collision. Because the momenta are at right angles to each other, we can use the Pythagorean theorem to find the resultant momentum: p_total = sqrt((p_car)^2 + (p_truck)^2) = sqrt((17000 kgm/s)^2 + (30000 kgm/s)^2) = 35000 kg*m/s.
Step 4: Calculate the angle of the total momentum. We can use trigonometry to find the angle. The tangent of the angle (θ) is equal to the opposite side (p_truck) divided by the adjacent side (p_car). So, tan(θ) = p_truck / p_car = 30000 kgm/s / 17000 kgm/s. To find the angle, we take the arctangent of this ratio: θ = arctan(30000/17000) = 60.26 degrees.
Step 5: Convert the angle to the east direction. The angle we calculated is relative to the west direction (the direction of the car's momentum). To convert this to the east direction, we subtract the angle from 180 degrees: 180 degrees - 60.26 degrees = 119.74 degrees.
So, the angle of their velocity after the collision relative to the east direction is approximately 119.74 degrees.
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