A police officer arrives at the scene of the collision of the two snowmobiles (Figure 1) to find both drivers unconscious. When the two vehicles collided, their skis became entangled and the two snowmobiles remained locked together as they skidded to a stop. One driver was thrown clear of the mishap, but the other driver remained in the driver¡¯s seat. The posted speed limit for snowmobiles in this cottage area is 60 km/h. The information the police officer obtained from eyewitness accounts and collision scene measurements are provided in Table 1. One witness described how driver A was thrown horizontally at a constant speed from his seat (0.5 m above the snow) surface to his final resting position.
a) Use the physics of kinematics, projectiles, conservation of momentum, and metric conversions to estimate the pre-collision speed of both vehicles.
Table 1
Mass of driver A
To estimate the pre-collision speed of both vehicles, we need to use the principles of kinematics, projectiles, conservation of momentum, and metric conversions. Let's break down the steps required to solve this problem:
1. Determine the final velocity of driver A: Given that driver A was thrown horizontally at a constant speed from his seat, we can assume that his final horizontal velocity is 0 m/s since he came to rest. (We are considering only the horizontal motion here because the vertical motion does not affect the final speed.)
2. Estimate the time taken by driver A to come to rest: The time of flight can be estimated using the equation for horizontal motion:
Horizontal distance = Horizontal velocity × Time
In this case, the horizontal distance is given as 0.5 m, and the horizontal velocity is unknown. Rearranging the equation, we get:
Time = Horizontal distance / Horizontal velocity
Since the time is the same for both drivers, we can use this equation for both drivers.
3. Calculate the initial velocity of each driver: Here, we need to apply the principles of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision (assuming no external force acts on the system). Mathematically, this can be expressed as:
Momentum before collision = Momentum after collision
The momentum of an object is given by the product of its mass and velocity (momentum = mass × velocity). Let's represent the mass of driver A as mA, the mass of driver B as mB, and the initial velocities of drivers A and B as VA and VB, respectively.
Momentum before collision = (mA × VA) + (mB × VB)
Momentum after collision = (mA × 0) + (mB × 0)
Since both drivers came to rest after the collision, their final velocities are both 0 m/s.
Setting the equations equal to each other, we get:
(mA × VA) + (mB × VB) = 0
Rearranging the equation, we can now solve for the initial velocity of each driver.
4. Convert the units: The given speed limit is 60 km/h, which needs to be converted to m/s for consistency. We can use the following conversion factor:
1 km/h = 1000 m/3600 s
Use this conversion factor to convert the speed limit to m/s.
By following these steps and plugging in the given information from Table 1, you should be able to estimate the pre-collision speed of both vehicles.