After you pick up a spare, your bowling ball rolls without slipping back toward the ball rack with a linear speed of v = 3.41 m/s
To reach the rack, the ball rolls up a ramp that rises through a vertical distance of h = 0.529 m. What is the linear speed of the ball when it reaches the top of the ramp?
To find the linear speed of the ball when it reaches the top of the ramp, we can use the principle of conservation of mechanical energy.
The mechanical energy of the ball is conserved in this case, so the initial mechanical energy equals the final mechanical energy. The initial mechanical energy is the sum of the kinetic energy and potential energy at the bottom of the ramp, while the final mechanical energy is the sum of the kinetic energy and potential energy at the top of the ramp.
At the bottom of the ramp, the ball has only kinetic energy since it is rolling without slipping, and potential energy is zero. Thus, the initial mechanical energy is equal to the kinetic energy, which is given by:
KE_initial = (1/2) * m * v^2
where m is the mass of the ball and v is the linear speed at the bottom of the ramp.
At the top of the ramp, the ball has both kinetic energy and potential energy. The kinetic energy is given by:
KE_final = (1/2) * m * v_final^2
where v_final is the linear speed of the ball at the top of the ramp.
The potential energy at the top of the ramp is given by:
PE_final = m * g * h
where g is the acceleration due to gravity and h is the vertical distance through which the ball rises.
Since the mechanical energy is conserved, we have:
KE_initial = KE_final + PE_final
Substituting the expressions for KE_initial, KE_final, and PE_final, we get:
(1/2) * m * v^2 = (1/2) * m * v_final^2 + m * g * h
Cancelling out the common factor of (1/2) * m, we have:
v^2 = v_final^2 + 2 * g * h
Rearranging the equation to solve for v_final, we get:
v_final^2 = v^2 - 2 * g * h
Finally, taking the square root of both sides, we find:
v_final = sqrt(v^2 - 2 * g * h)
Now we can plug in the values: v = 3.41 m/s, h = 0.529 m, and g ≈ 9.8 m/s^2 to calculate the linear speed when the ball reaches the top of the ramp.