Respuesta :
Answer:
(-2.32, 2.63) kg m/s
Explanation:
From Newton's second law of motion, force is the rate of change of momentum. Mathematically,
[tex]F = \dfrac{dp}{dt}[/tex]
where p is the momentum.
Integrating both sides,
[tex]p = \int\!F\, dt + C[/tex]
where C is a constant of integration determined by initial values.
[tex]p = \int\!(0.275t\,\hat{i} - 0.460t^2\,\hat{j}) dt + C[/tex]
[tex]p = \dfrac{0.275}{2}t^2\,\hat{i} - \dfrac{0.460}{3}t^3\,\hat{j} + C[/tex]
The initial momentum is when t = 0.
[tex]-2.90\,\hat{i}+3.95\,\hat{j} = \dfrac{0.275}{2}0^2\,\hat{i} - \dfrac{0.460}{3}0^3\,\hat{j} + C[/tex]
[tex]C = -2.90\,\hat{i}+3.95\,\hat{j}[/tex]
Hence,
[tex]p = \dfrac{0.275}{2}t^2\,\hat{i} - \dfrac{0.460}{3}t^3\,\hat{j} -2.90\,\hat{i}+3.95\,\hat{j} = (\dfrac{0.275}{2}t^2-2,90)\,\hat{i} + (3.95-\dfrac{0.460}{3}t^3)\,\hat{j}[/tex]
At t = 2.05 s,
[tex]p = (\dfrac{0.275}{2}(2.05)^2-2,90)\,\hat{i} + (3.95-\dfrac{0.460}{3}(2.05)^3)\,\hat{j}[/tex]
[tex]p = -2.32\,\hat{i} + 2.63\,\hat{j}[/tex]
Thus, p = (-2.32, 2.63) kg m/s
The x and y components of the momentum are -2.32 and 0.63 kg m/s respectively for the box at 2.05 s.
According to the second law of motion, force is the rate of change of momentum.
[tex]\bold {F = \dfrac {dp}{dt}}[/tex]
Where,
p - momentum
Integrate both sides,
[tex]\bold {p = \int\limits F \, dt+C }[/tex]
Where,
C - constant of integration
[tex]\bold {p = \int\limits {0.275t\ \hat {i} - 0.0460t^2 \hat j )} \, dt+C }\\\\\bold {p = (\dfrac {0.275t^2}{2}\ \hat {i} - \dfrac {0.0460}{3}t^2 \hat j ) \, dt+C }[/tex]
For initial momentum t = 0
[tex]\bold { (-2.90)\hat i+(3.95kg)\hat j =\bold (\dfrac {0.275t^2}{2}\ \hat {i} - \dfrac {0.0460}{3}t^2 \hat j ) \, dt+C } }\\\\\bold {C = (-2.90)\hat i+(3.95kg)\hat j }[/tex]
Thus, momentum at t= 2.5 sec,
[tex]\bold {p = (\dfrac {0.275\times (2.05)^2}{2}\ \hat {i} - \dfrac {0.0460}{3}(0.25)^3 \hat j ) \, dt+C }\\\\\bold {p = {- 2.32 \hat {i} - 0.63 \hat j )} }\\\\\bold {p = {- 2.32, 0.63\ kg.m/s )} }\\[/tex]
The x and y components of the momentum are -2.32 and 0.63 kg m/s respectively for the box at 2.05 s.
To know more about momentum,
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