Two automobiles of equal mass approach an intersection. One vehicle is traveling with velocity 15.5 m/s toward the east and the other is traveling north with speed v2i. Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 51.0° north of east. What was the initial speed of the northward-moving vehicle?
To find the initial speed of the northward-moving vehicle, we can use the principle of conservation of momentum.
1. Identify the given information:
- Velocity of the eastward-moving vehicle (v1) = 15.5 m/s
- Angle between the skid marks (θ) = 51.0°
2. Break down the given information:
- The eastward-moving vehicle has a velocity component in the x-axis direction (v1x) and no velocity component in the y-axis direction (v1y).
- The northward-moving vehicle has a velocity component in the y-axis direction (v2y) and no velocity component in the x-axis direction (v2x).
3. Convert the angle to radians:
- Angle in radians (θ_radians) = θ * (π/180)
4. Determine the components of velocity for each vehicle:
- The x-component of the eastward-moving vehicle (v1x) is given by v1x = v1 * cos(0°) = v1
- The y-component of the northward-moving vehicle (v2y) is given by v2y = v2 * sin(90°) = v2
5. Apply the conservation of momentum equation:
- Net momentum before the collision = Net momentum after the collision
- (m1 * v1x) + (m2 * v2x) = (m * vf)
- Since the vehicles stick together, their combined mass (m) and final velocity (vf) are the same for both vehicles.
6. Solve for the initial speed of the northward-moving vehicle (v2):
Step 1: Calculate the x-component of the net momentum after the collision using the final velocity (vf) and the given angle (θ_radians):
- Net momentum after the collision in the x-axis direction (m * vf * cos(θ_radians)) = vf * cos(θ_radians)
Step 2: Substitute the values and solve for v2:
- (m1 * v1) + 0 = m * vf * cos(θ_radians)
- v2 = (m1 * v1) / (m * cos(θ_radians))
7. Calculate the initial speed of the northward-moving vehicle (v2):
- Substitute the values of m1, v1, m, and θ_radians in the equation from the previous step.
Remember to convert any angles to radians and double-check if the units are consistent throughout the calculation.