Question

A boy hits a baseball with a bat and imparts an impulse J to the ball. The boy hits the ball again with the same force, except that the ball and the bat are in contact for twice the amount of time as in the first hit. The new impulse equals:
A) Half the original impulse
B) The original impulse
C) Twice the original impulse
D) Four times the original impulse

Answer 1

One can solve this question instantly if he/she is well aware of the concept of impulse. The term impulse implies the effect of a force acting on a body over time to cause a change in momentum of the body. If “F” is a force acting on a body over the time “$\Delta$ T” so the impulse “J” is given by, ${J} ={{F} \times {\Delta T}}$

Now if the force is constant then one can say that impulse is directly proportional to the time period hence, the impulse will increase proportionally with the increase in time. I.e. If time is doubled then the impulse will also be doubled.

Complete step by step answer:
We know that,
${J} = {{F} \times {\Delta T}}$
Hence, impulse for the first case is given by,
${J_1} = {{F} \times {\Delta {T_1}}}$………. (1)
Similarly, impulse for the second case can be written as,
${J_2} = {{F} \times {\Delta {T_2}}}$………. (2)
But, the time in second case is twice the time in first case, hence mathematically we can write
$\Delta {T_2} = 2\Delta {T_1}$,
putting this value in equation (2) we get,
${J_2} = {{F} \times {(2\Delta {T_1})}}$
${J_2} = 2({F} \times {\Delta {T_1}})$
${J_2} = 2{J_1}$………. (from equation 1)

Hence, option C is the right choice i.e. the new impulse will be twice that of the original impulse.

Note: Generally, the terms impulse and momentum are used together so one may get confused between the terminologies so it becomes essential to know the difference between these terms. Impulse is a term that describes the effect of force over time to cause a change in momentum of a body. Momentum is usually defined as the product of mass and velocity, which is nothing but a measure of some amount of mass being in motion.