When a goalkeeper stops the ball he does the work but how he only stops the ball but dont displaces the ball please explain
Answer
Does the goalkeeper do Work while stopping the ball?
The amount of Work done by the goalkeeper’s mass equals the amount of kinetic energy lost by the ball.
Computing the velocity of the mass-plus-ball system after the goalkeeper captures the ball gives us a quantitative answer as to the amount of Work done by the goalkeeper’s mass.
To solve this problem, assume the goalkeeper captures the ball in a frictionless process, (e.g., the ball sticks to the goalkeeper, and the goalkeeper begins to slide on a frictionless surface). The goalkeeper gains kinetic energy, and the ball loses kinetic energy. The total amount of kinetic energy in the ball before the capture is the same as the kinetic energy of the ball-goalkeeper system after the capture.
The big picture: the goalkeeper’s mass exerts an inertial force on the ball, causing the ball to decelerate. Likewise, the ball exerts an inertial force on the goalkeeper, causing him/her to accelerate. The amount of kinetic energy lost by the ball is the amount of Work done by the inertial force exerted by the ball on the mass of the goalkeeper.
The calculation is performed by equating the kinetic energy of the ball before the collision, with the kinetic energy of the goalkeeper-plus-ball after the collision, and then solving for the velocity.
If we choose the goalkeeper at rest as the $(0,0)$ origin of the frame of our system, in this frame, the goalkeeper has zero kinetic energy. The ball has velocity $v_i$ and mass $m_{b}$, and therefore $KE=\frac{1}{2}m_{b}v_i^2$.
After the goalkeeper catches the ball, the goalie plus ball system will have velocity $v_f$. The system of the ball-goalkeeper will have the same total kinetic energy as was present in the ball before the collision. The transfer of energy (and Work done) is obvious in hockey when the goalkeeper is on frictionless ice. After catching the ball, both goalkeeper and puck move backward with a lower velocity than the puck. The work done by the mass of the goalkeeper on the puck/ball is equal to the change of kinetic energy of the puck/ball. $$W=\Delta KE = \frac {1}{2} m_b (v_i -v_f) ^2$$
The key to understanding this concept is realizing that mass exerts an inertial force on any body attempting to accelerate that mass. Thus, the total Work done in the process can be calculated by integrating the force acting over each increment of distance moved. $W=\int F dx$. Obviously, this is a difficult way to calculate the Work done by the ball on the goalkeeper because the force exerted by the ball changes each moment over a very short period of time. We would need equipment to measure the force vs. time function to do the calculation. The easier method is to simply measure the speed of the ball before capture, and speed of the goalkeeper-ball system afterward, and calculate the change in kinetic energy of the ball.
In the process of the collision (catching the ball), the goalkeeper accelerates, and the ball decelerates, which means the ball exerts an inertial force on the goalkeeper, and the goalkeeper exerts an inertial force on the ball.
The inertial force exerted by a mass is the force the mass exerts against a force attempting to accelerate that mass. Note: the inertial force exerted by a mass will always be opposite and equal to the force applied to the mass. For example: The goalkeeper feels the inertial force of the ball when it hits his/her hands. As the ball contacts the goalkeeper’s hands, the force applied on his/her hands is opposite and equal to the force applied by his/her hands on the ball. But, because of the conservation of energy and momentum (and a lot more discussion of what that means), even though the forces are opposite and equal, the ball decelerates, and the mass of the ball-goalkeeper system accelerates.
In short, the inertial force exerted by the ball on the goalkeeper is the force referred to in Newton's second law, $F_{inertial}=ma$.
Work, in its most general sense, is defined as the amount of energy transferred into or out of a system. If energy leaves a system, we typically refer to that transaction as the system “doing work,” and vice versa. In the case of the goalkeeper, the ball did work on the goalkeeper by accelerating him/her. The perspective of "doing Work on," or "having Work done to" a system, changes depending on the frame chosen and system boundaries.
In this question, we see the goalkeeper is accelerated, and the ball is decelerated, and so energy is transferred, and hence Work is done.
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