You drop a ball from a height of 2.8 m, and it bounces back to a height of 1.4 m. (Ignore air resistance.)
(a) What fraction of its initial energy is lost during the bounce?
(b) What is the ball's speed just as it leaves the ground after the bounce?
m/s
(c) Where did the energy go?
To answer these questions, we can use the principle of conservation of energy. The potential energy of an object at a certain height can be calculated using the formula PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
(a) To determine the fraction of energy lost during the bounce, we compare the potential energy before and after the bounce. The initial potential energy of the ball is PE_initial = m * g * h_initial, where h_initial is the initial height of 2.8 m.
The potential energy after the bounce is PE_final = m * g * h_final, where h_final is the height after the bounce, which is 1.4 m.
The fraction of energy lost can be calculated by taking the difference between the initial and final potential energy and dividing it by the initial potential energy, and then multiplying by 100 to express it as a percentage.
Fraction of energy lost = [(PE_initial - PE_final) / PE_initial] * 100
(b) To find the speed of the ball just as it leaves the ground after the bounce, we can use the principle of conservation of energy. The potential energy at the maximum height of 1.4 m is converted entirely into kinetic energy just as the ball leaves the ground.
The total energy just as the ball leaves the ground is the sum of potential and kinetic energy. As the ball reached the maximum height of 1.4 m, all potential energy is transformed into kinetic energy, so we can write:
Potential energy = Kinetic energy
m * g * h = (1/2) * m * v^2
Where v is the velocity or speed of the ball just as it leaves the ground. We can solve this equation for v:
v = sqrt(2 * g * h)
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
(c) The energy lost during the bounce is typically transferred into other forms of energy, such as heat and sound. When the ball hits the ground, some of its kinetic energy is converted into internal energy of the ball and the surface it hits. This internal energy is usually dissipated as heat and sound energy.