Betty Bodycheck (mB = 51.4 kg, vB = 22.0 km/h in the positive x-direction) and Sally Slasher (mS = 42.4 kg, vS = 28.0 km/h in the positive y-direction) are both racing to get to a hockey puck. Immediately after the collision, Betty is heading in a direction that is 76.0° counterclockwise from her original direction, and Sally is heading back and to her right in a direction that is 12.0° from the x-axis.
What are Betty and Sally's final kinetic energies?
To find Betty and Sally's final kinetic energies, we need to use the conservation of momentum and the conservation of mechanical energy.
1. Conservation of Momentum:
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. The momentum is given by the product of mass and velocity.
Before the collision:
Betty's momentum (pB) = mB * vB
Sally's momentum (pS) = mS * vS
After the collision:
The magnitude of Betty's momentum after the collision (pB') can be found using the principle of vector addition:
pB' = √((pxB')^2 + (pyB')^2)
Similarly, the magnitude of Sally's momentum after the collision (pS') can be found using the same principle:
pS' = √((pxS')^2 + (pyS')^2)
2. Conservation of Mechanical Energy:
The mechanical energy before the collision is equal to the mechanical energy after the collision. Mechanical energy is the sum of kinetic energy given by 1/2 * mass * velocity^2.
Before the collision:
Betty's kinetic energy (KB) = 1/2 * mB * vB^2
Sally's kinetic energy (KS) = 1/2 * mS * vS^2
After the collision:
The magnitude of Betty's kinetic energy after the collision (KB') can be found using the formula:
KB' = 1/2 * mB * vB'^2
Similarly, the magnitude of Sally's kinetic energy after the collision (KS') can be found using the formula:
KS' = 1/2 * mS * vS'^2
Now we will substitute the given values and solve step by step.
Given values:
mB = 51.4 kg (mass of Betty)
vB = 22.0 km/h (velocity of Betty)
mS = 42.4 kg (mass of Sally)
vS = 28.0 km/h (velocity of Sally)
To find the final velocities, we need to convert the velocities from km/h to m/s:
vB = 22.0 km/h * (1000 m/3600 s) = 6.11 m/s (velocity of Betty)
vS = 28.0 km/h * (1000 m/3600 s) = 7.78 m/s (velocity of Sally)
Calculate the momenta before the collision:
pB = mB * vB = 51.4 kg * 6.11 m/s = 313.854 kg·m/s (momentum of Betty)
pS = mS * vS = 42.4 kg * 7.78 m/s = 329.872 kg·m/s (momentum of Sally)
Now let's consider the momenta after the collision:
The x-component of Betty's momentum after the collision (pxB') can be obtained using the magnitude and direction:
pxB' = pB' * cos(76°)
The y-component of Sally's momentum after the collision (pyB') can be obtained using the magnitude and direction:
pyB' = pB' * sin(76°)
The x-component of Sally's momentum after the collision (pxS') can be obtained using the magnitude and direction:
pxS' = pS' * cos(12°)
The y-component of Sally's momentum after the collision (pyS') can be obtained using the magnitude and direction:
pyS' = pS' * sin(12°)
To calculate the kinetic energies, we need to find the final velocities.
The magnitude of Betty's velocity after the collision (vB') can be obtained using the x and y-components of the momentum:
vB' = √((pxB')^2 + (pyB')^2)
The magnitude of Sally's velocity after the collision (vS') can be obtained using the x and y-components of the momentum:
vS' = √((pxS')^2 + (pyS')^2)
Finally, we can calculate the final kinetic energies using the formulas:
KB' = 1/2 * mB * vB'^2
KS' = 1/2 * mS * vS'^2
Substitute the values and calculate KB' and KS' to get Betty and Sally's final kinetic energies.