An impulse of [tex]\boxed{345\text{ N}\cdot\text{s}}[/tex] to the west is required to decrease the speed of the boat from [tex]13\text{ m/s}[/tex] east to [tex]10\text{ m/s}[/tex] east.
Further explanation:
When a force acted upon an object for a very small time, is called impulse. Force is a rate of change of momentum with respect to time. Therefore if momentum associated with an object is changed in a very short time, the force exerted on the object is called the impulse.
Given:
Mass of the boat is [tex]115\text{ kg}[/tex].
Initial speed of the boat is [tex]13\text{ m/s}[/tex] to the east.
Final velocity of the boat is [tex]10\text{ m/s}[/tex] to the east.
Concept:
Let’s assume east direction as the positive direction of x axis.
Write the expression for impulse
[tex]\begin{aligned}I&={{\vec F}_{{\text{avg}}}}\cdot\Delta t\\&=\Delta m\vec V\\\end{aligned}[/tex]
Here, [tex]I[/tex] is the impulse force, [tex]\Delta m\vec V[/tex] is the change of momentum, [tex]\Delta t[/tex] is the time and [tex]m[/tex] is the mass of the boat.
In the above expression [tex]m[/tex] is a constant, therefore above expression can be rewritten as
[tex]I=m\cdot\Delta\vec V[/tex] …… (1)
Here, [tex]\Delta\vec V[/tex] is the change in the velocity of the boat.
Write the expression for change in velocity
[tex]\Delta\vec V={V_{\text{f}}}-{V_{\text{i}}}[/tex] …… (2)
Here, [tex]{V_{\text{f}}}[/tex] is the final velocity of boat and [tex]{V_{\text{i}}}[/tex] is the initial velocity of the boat.
Substitute [tex]13\text{ m/s}[/tex] for [tex]{V_{\text{i}}}[/tex] and [tex]10\text{ m/s}[/tex] for [tex]{V_{\text{f}}}[/tex] in equation (2).
[tex]\begin{aligned}\Delta \vec V&=\left( {10\,\hat i}\right)-\left( {13\,\hat i}\right)\\&=3\,\left({-\hat i}\right)\\\end{aligned}[/tex]
Substitute [tex]3\,{\text{m/s}}\,\left( { - \hat i} \right)[/tex] for [tex]\Delta \vec V[/tex] and [tex]115\text{ kg}[/tex] for [tex]m[/tex] in equation (1).
[tex]\begin{aligned}I&=\left({115\,{\text{kg}}}\right)\left( {-3\,{\text{m/s}}\,\hat i}\right)\\&=345\,{\text{N.s}}\left({-\hat i} \right)\\\end{aligned}[/tex]
Here, [tex]-I[/tex] indicate that the impulse acts in opposite of east direction.
Thus, an impulse of [tex]\boxed{345\text{ N.s}}[/tex] to the west is required to decrease the speed of the boat from [tex]13\text{ m/s}[/tex] east to [tex]10\text{ m/s}[/tex] east.
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Answer Details:
Grade: High School
Subject: Physics
Chapter: Force
Keywords:
Boat, mass, 115 kg, drifts, river, 13 m/s, 13 meter per second, east, impulse, decrease, speed, 10 m/s, 10 meter per second, 345 Ns, west, 3.45 times 10^2 Ns.
