A bowling ball encounters a 0.760-m vertical rise on the way back to the ball rack, as the drawing illustrates. Ignore frictional losses and assume that the mass of the ball is distributed uniformly. The translational speed of the ball is 7.83 m/s at the bottom of the rise. Find the translational speed at the top.
0.7
To find the translational speed at the top of the rise, we can use the principle of conservation of energy. The total mechanical energy of the bowling ball is conserved, assuming there is no friction.
The mechanical energy of an object is given by the sum of its kinetic energy and potential energy. At the bottom of the rise, the ball only has kinetic energy, and at the top, it only has potential energy.
At the bottom of the rise:
Kinetic energy = 1/2 * mass * velocity^2
At the top of the rise:
Potential energy = mass * gravity * height
Since the mass of the ball is distributed uniformly, we can rewrite the potential energy equation as:
Potential energy = mass * gravity * (height_bottom + height_change)
Given:
velocity_bottom = 7.83 m/s
height_change = 0.760 m
We need to find the velocity at the top, which is the translational speed.
1. Calculate the mass of the bowling ball:
The mass of the ball is not given in the problem statement. We can leave it as a variable, denoted as m.
2. Calculate the kinetic energy at the bottom:
Kinetic energy_bottom = 1/2 * m * velocity_bottom^2
3. Calculate the potential energy at the top:
Potential energy_top = m * gravity * (height_bottom + height_change)
4. Set the kinetic energy at the bottom equal to the potential energy at the top:
1/2 * m * velocity_bottom^2 = m * gravity * (height_bottom + height_change)
5. Solve for the translational speed at the top:
velocity_top = sqrt((2 * gravity * (height_bottom + height_change) * velocity_bottom^2) / (velocity_bottom^2 + 2 * gravity * (height_bottom + height_change)))
Substituting the given values into the formula and solving the equation will give you the translational speed at the top of the rise.
To find the translational speed of the ball at the top of the rise, we can apply the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system, consisting of kinetic energy (KE) and potential energy (PE), remains constant if there are no external forces acting on the system.
The mechanical energy of the bowling ball at the bottom of the rise (Point A) is the sum of its kinetic energy and potential energy:
E_A = KE_A + PE_A
The mechanical energy of the ball at the top of the rise (Point B) is:
E_B = KE_B + PE_B
Since the problem statement mentions that there are no frictional losses, the only forces acting on the ball are gravitational forces and normal forces. The potential energy at both points can be calculated using the equation:
PE = mgh
where m is the mass of the ball, g is the acceleration due to gravity, and h is the height.
At the bottom of the rise (Point A), the ball has a translational speed of 7.83 m/s. Therefore, its kinetic energy can be calculated using the equation:
KE = 0.5mv^2
where m is the mass of the ball and v is the speed.
To find the translational speed at the top of the rise (Point B), we need to equate the mechanical energies at points A and B, i.e., E_A = E_B:
KE_A + PE_A = KE_B + PE_B
Substituting the equations for potential energy and kinetic energy, we have:
(0.5m(7.83)^2) + mgh_A = 0.5mv_B^2 + mgh_B
Simplifying the equation and rearranging to solve for v_B, we get:
v_B = sqrt((v_A)^2 + 2gh_A - 2gh_B)
where v_A is the translational speed at the bottom of the rise, h_A is the height at the bottom, and h_B is the height at the top.
By substituting the given values of h_A, h_B, and v_A into the equation, you can calculate the translational speed of the ball at the top of the rise.