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The impulse of a superball and a blob of clay can be obtained when the door exerts a force for a very small time. There is a change in the momentum of the superball and a blob of clay. Thus, with the help of this, the impulse in both cases can be determined.

The blob of clay is thrown on the door. Then, the clay will stick to the door. The door will exert an impulse equal to the change in momentum of a blob of clay. Clay will stick with the door and move along with the door. It will transfer its energy to the door, which will ultimately close the door.

The momentum of the clay blob is equal to the impulse exerted by the door.

An impulse is equal to the change in the momentum of the clay blob. The impulse can be calculated as

\(I = m\left( {{v_f} - {v_i}} \right)\).

Here,\({v_f}\)is the final velocity of the blob of clay, which is equal to zero;\({v_i}\)is the initial velocity of the clay; m is the mass of the blob of clay, which is also the same as the superball.

Substitute the values in the above equation.

\(\begin{aligned}{c}I = m\left( {0 - {v_i}} \right)\\ = - m{v_i}\end{aligned}\)

Here, the negative sign indicates the opposite direction of velocity.

The magnitude of the impulse is equal to the value of the initial momentum of the blob of clay.

The superball rebounds back in the opposite direction, with the same momentum. But during the rebound, the momentum acts in an opposite direction of the initial momentum during the hitting with the door. There is a negligible loss in the speed of the superball mass.

An impulse of the system is equal to the change in the momentum of the system. The impulse can be calculated as

\({I_2} = m\left( {{v_{{f_2}}} - {v_{{i_2}}}} \right)\).

Here,\({v_{{f_2}}}\)is the final velocity of the superball is equal to\(\left( { - {v_{{i_2}}}} \right)\);\({v_{{i_2}}}\)is the initial velocity of the superball; m is the mass of the superball.

Substitute the values in the above equation.

\(\begin{aligned}{c}{I_2} = m\left( { - {v_{{i_2}}} - {v_{{i_2}}}} \right)\\ = - 2m{v_{{i_2}}}\end{aligned}\)

Here, the negative sign indicates the opposite direction of velocity.

An impulse force exerted by the door in the opposite direction is equal to two times the momentum of the superball.

The momentum of the superball is equal to twice the value of the initial momentum; in the case of the superball, the magnitude of impulse increases. Thus, the superball will be more effective to throw at the door.

Option (a) can be the correct option because the momentum of the superball increases, thus increasing the values of impulse.

Option (b) is incorrect because the blob of clay impulse is less in magnitude than the superball.

Option (c) is incorrect because the magnitude of an impulse is not the same in both cases.

Option (d) is incorrect because superball will work more effectively in closing the door.

Hence, from the above analysis of options, the superball would be more effective to throw at the door for its closing.

Thus, option (a) is correct.