Two spheres of equal mass are thrown against a wall with equal velocities. One sphere rebounds with 75% of its original speed, while the other sticks to the wall. Which imparted a greater impulse to the wall, the sticky ball or the rebounding ball? Justify your answer.
To determine which sphere imparted a greater impulse to the wall, we need to understand the concept of impulse. Impulse is defined as the change in momentum of an object, and it is calculated by multiplying the force applied to an object by the time interval over which the force is applied.
In this scenario, both spheres have the same mass and are thrown with equal velocities, so their initial momentum is the same. Let's assume the initial momentum of each sphere is P.
For the sphere that rebounds, its final velocity is 75% of its initial velocity. Let's call this final velocity Vr. So, the final momentum of the rebounding ball is final momentum (Pr) = mass * Vr = m * (0.75 * initial velocity) = 0.75mV.
For the sphere that sticks to the wall, its final velocity is 0 since it comes to rest. The final momentum of the sticky ball, Ps, is zero.
The impulse imparted to the wall by each sphere can be calculated using the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event.
Initial momentum before the collision = Sum of initial momenta of both spheres = P + P = 2P.
Final momentum after the collision = Sum of final momenta of the spheres = Pr + Ps = 0.75mV + 0.
According to the conservation of momentum, the initial momentum should be equal to the final momentum. Therefore, 2P = 0.75mV.
Now, let's compare the magnitudes of the impulses for each scenario. The impulse (J) is given by the change in momentum: J = Final momentum - Initial momentum.
For the rebounding sphere, the impulse is Jr = Pr - P = 0.75mV - P.
For the sticky sphere, the impulse is Js = Ps - P = 0 - P.
Comparing the two impulses, Jr - Js = (0.75mV - P) - (0 - P) = 0.75mV.
Since the magnitude of Jr (0.75mV) is greater than Js (0), the rebounding ball imparts a greater impulse to the wall.
Therefore, the rebounding ball imparts a greater impulse to the wall compared to the sticky ball.