Question

There are (one can say) three coequal theories of motion for a single particle: Newton’s second law, stating that the total force on the particle causes its acceleration; the work-kinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulse-momentum theorem, stating that the total impulse on the particle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A $3.00{\text{ }}Kg$ object has velocity $7.00{{ \hat j m}}{{\text{s}}^{ - 1}}$. Then a constant net force $12.0{{ \hat i N}}$ acts on the object for 5.00 s. Calculate its acceleration from \[\vec a = \dfrac{{({{\vec v}_f} - {{\vec v}_i})}}{{\Delta t}}\].

Answer 1

Hint: In order to solve the question, we will first of all use the impulse-momentum theorem so as to find the final velocity of the object after we find the final velocity we will use the formula given in the question to find the acceleration of the object for which we will use the initial velocity and time of the object given in the question.

Formula used:
The impulse-momentum theorem
${P_f} - {P_i} = F \times t$
Basic formula of acceleration
\[\vec a = \dfrac{{({{\vec v}_f} - {{\vec v}_i})}}{{\Delta t}}\]
Here, $F$= force, $m$= mass, $a$= acceleration, ${P_i}$ = initial momentum, ${P_f}$ = final momentum, $t$ = time, ${v_i}$ = initial velocity and ${v_f}$ = final velocity.

Complete step by step answer:
In the question we are given three theories of motion for a single particle: Newton’s second law, stating that the total force on the particle causes its acceleration. The work-kinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy.

The impulse-momentum theorem, stating that the total impulse on the particle causes its change in momentum
Mass of object = $3.00{\text{ }}Kg$
Velocity of object = $7.00{{ \hat j m}}{{\text{s}}^{ - 1}}$
Constant net force applied on the object = $12.0{{ \hat i N}}$
time taken = 5.00 s.
To use the given formula we need to find $\Delta {v_f}$.According to The impulse-momentum theorem,
${P_f} - {P_i} = F \times t$
\[\text{change in momentum} = \text{force} \times \text{time}\]

Putting the values according to question
${P_i} + F \times t = {P_f}$
$\Rightarrow m{v_i} + F \times t = m{v_f}$
$\Rightarrow (3.00{\text{ }}kg) \times (7.00{{ \hat j m}}{{\text{s}}^{ - 1}}) + (12.0{{ \hat i N}}) \times 5.00s = (3.00{\text{ }}kg) \times {v_f}$
After solving the equation, we get
\[{v_f} = (20.0{{ \hat i + }}7.00{{ \hat j )m}}{{\text{s}}^{ - 1}}\]
Now putting the value in given formula
\[\vec a = \dfrac{{({{\vec v}_f} - {{\vec v}_i})}}{{\Delta t}}\]
\[\Rightarrow \vec a = \dfrac{{(20.0{{ \hat i + }}7.00{{ \hat j - }}7.00{{ \hat j )}}}}{{5.00}}{\text{ m}}{{\text{s}}^{ - 1}}\]
Solving the equation, we get
\[\therefore \vec a = 4.00{{ \hat i m}}{{\text{s}}^{ - 1}}\]

Hence, the correct option is acceleration of the object is \[\vec a = 4.00{{ \hat i m}}{{\text{s}}^{ - 1}}\].

Note: Many of the students will make the mistake by not giving the answer in significant figure and in vector direction in the question we are given all the physical quantities in three significant figure so the answer should be in that only along with this force and velocity is given in vectors to acceleration should be in answer we cannot write only the magnitude.