Betty Bodycheck (mB = 55.1 kg, vB = 22.0 km/h in the positive x-direction) and Sally Slasher (mS = 41.2 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?
Betty's final kinetic energy = 602.2 J
Sally's final kinetic energy = 517.2 J
To find Betty and Sally's final kinetic energies, we need to calculate their velocities after the collision using the given information and then use the formula for kinetic energy.
Let's break down the problem into steps:
Step 1: Convert Betty and Sally's initial velocities from km/h to m/s:
vB = 22.0 km/h = 22.0 m/s (since 1 km/h = 1,000 m/3,600 s, dividing km/h by 3.6 gives m/s)
vS = 28.0 km/h = 28.0 m/s
Step 2: Calculate Betty's final velocity using the given direction:
Betty is heading in a direction that is 76.0° counterclockwise from her original direction. We can represent her final velocity as a vector with x- and y-components.
The x-component of Betty's final velocity, vBx', can be found using the formula:
vBx' = vB * cos(theta)
where theta is the counterclockwise angle from the x-axis (76.0°).
vBx' = 22.0 m/s * cos(76.0°)
Step 3: Calculate Sally's final velocity using the given direction:
Sally is heading back and to her right in a direction that is 12.0° from the x-axis. We can also represent her final velocity as a vector with x- and y-components.
The x-component of Sally's final velocity, vSx', can be found using the formula:
vSx' = vS * cos(theta)
where theta is the angle from the x-axis (12.0°).
vSx' = 28.0 m/s * cos(12.0°)
Step 4: Calculate Betty and Sally's final kinetic energies using the formula:
Kinetic energy (K) = (1/2) * mass * velocity^2
The final kinetic energy for Betty (KB') is given by:
KB' = (1/2) * mB * (vBx')^2
The final kinetic energy for Sally (KS') is given by:
KS' = (1/2) * mS * (vSx')^2
Let's calculate the final kinetic energies:
Step 1: Convert the given angles to radians:
76.0° = 76.0 * (π/180) ≈ 1.326 rad
12.0° = 12.0 * (π/180) ≈ 0.209 rad
Step 2: Calculate Betty's final x-component velocity (vBx'):
vBx' = 22.0 m/s * cos(1.326 rad)
Step 3: Calculate Sally's final x-component velocity (vSx'):
vSx' = 28.0 m/s * cos(0.209 rad)
Step 4: Calculate the final kinetic energies:
KB' = (1/2) * 55.1 kg * (vBx')^2
KS' = (1/2) * 41.2 kg * (vSx')^2
By substituting the calculated values into the equations, you will find the final kinetic energies of Betty and Sally.
To find Betty and Sally's final kinetic energies, we need to calculate their final velocities first.
We can start by breaking down Betty and Sally's initial velocities into their x and y components:
For Betty:
mB = 55.1 kg
vB = 22.0 km/h in the positive x-direction
Using basic trigonometry, we can find the x-component of Betty's velocity:
vBx = vB * cos(0°) = 22.0 km/h * cos(0°) = 22.0 km/h
And for the y-component:
vBy = vB * sin(0°) = 22.0 km/h * sin(0°) = 0 km/h
So, Betty's initial velocity can be expressed as:
vB = (vBx, vBy) = (22.0 km/h, 0 km/h)
Similarly, for Sally:
mS = 41.2 kg
vS = 28.0 km/h in the positive y-direction
The x-component of Sally's velocity is:
vSx = vS * cos(90°) = 28.0 km/h * cos(90°) = 0 km/h
And the y-component can be found as:
vSy = vS * sin(90°) = 28.0 km/h * sin(90°) = 28.0 km/h
So, Sally's initial velocity is:
vS = (vSx, vSy) = (0 km/h, 28.0 km/h)
Now, let's calculate the final velocities after the collision.
After the collision, Betty is heading in a direction that is 76.0° counterclockwise from her original direction. This means her new velocity has a magnitude but a different direction.
To find Betty's new x and y components, we use the following formulas:
vBfx = |vBf| * cos(76.0°)
vBfy = |vBf| * sin(76.0°)
Similarly, Sally is heading back and to her right in a direction that is 12.0° from the x-axis.
vSfx = |vSf| * cos(12.0°)
vSfy = |vSf| * sin(12.0°)
Now, we need to use the principles of conservation of momentum and conservation of kinetic energy to find the velocities.
Conservation of momentum states that the sum of initial momenta is equal to the sum of final momenta:
mB * vB = mB * vBf + mS * vSf
Substituting the known values:
55.1 kg * (22.0 km/h) = 55.1 kg * vBfx + 41.2 kg * vSfx --> Equation 1
Similarly, for the y-components:
0 kg * (0 km/h) + 41.2 kg * (28.0 km/h) = 55.1 kg * vBfy + 41.2 kg * vSfy --> Equation 2
Conservation of kinetic energy states that the sum of initial kinetic energies is equal to the sum of the final kinetic energies:
(1/2) * mB * (vB)^2 + (1/2) * mS * (vS)^2 = (1/2) * mB * (vBf)^2 + (1/2) * mS * (vSf)^2
Substituting the known values:
(1/2) * 55.1 kg * (22.0 km/h)^2 + (1/2) * 41.2 kg * (28.0 km/h)^2 = (1/2) * 55.1 kg * (vBfx)^2 + (1/2) * 41.2 kg * (vSfx)^2 + (1/2) * 55.1 kg * (vBfy)^2 + (1/2) * 41.2 kg * (vSfy)^2
Now, we need to solve these equations to find vBfx, vBfy, vSfx, and vSfy.
Once we have the final velocities, we can calculate the final kinetic energies using the formula:
Kf = (1/2) * m * (vf)^2
where Kf is the final kinetic energy, m is the mass, and vf is the final velocity.
However, since we don't have the exact initial velocities and angles in this question, it is not possible to determine the final velocities and kinetic energies without additional information.