An impulse of \(\boxed{345\text{ N}\cdot\text{s}}\) to the west is required to decrease the speed of the boat from \(13\text{ m/s}\) east to \(10\text{ m/s}\) 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 \(115\text{ kg}\).
Initial speed of the boat is \(13\text{ m/s}\) to the east.
Final velocity of the boat is \(10\text{ m/s}\) to the east.
Concept:
Let’s assume east direction as the positive direction of x axis.
Write the expression for impulse
\(\begin{aligned}I&={{\vec F}_{{\text{avg}}}}\cdot\Delta t\\&=\Delta m\vec V\\\end{aligned}\)
Here, \(I\) is the impulse force, \(\Delta m\vec V\) is the change of momentum, \(\Delta t\) is the time and \(m\) is the mass of the boat.
In the above expression \(m\) is a constant, therefore above expression can be rewritten as
\(I=m\cdot\Delta\vec V\) …… (1)
Here, \(\Delta\vec V\) is the change in the velocity of the boat.
Write the expression for change in velocity
\(\Delta\vec V={V_{\text{f}}}-{V_{\text{i}}}\) …… (2)
Here, \({V_{\text{f}}}\) is the final velocity of boat and \({V_{\text{i}}}\) is the initial velocity of the boat.
Substitute \(13\text{ m/s}\) for \({V_{\text{i}}}\) and \(10\text{ m/s}\) for \({V_{\text{f}}}\) in equation (2).
\(\begin{aligned}\Delta \vec V&=\left( {10\,\hat i}\right)-\left( {13\,\hat i}\right)\\&=3\,\left({-\hat i}\right)\\\end{aligned}\)
Substitute \(3\,{\text{m/s}}\,\left( { - \hat i} \right)\) for \(\Delta \vec V\) and \(115\text{ kg}\) for \(m\) in equation (1).
\(\begin{aligned}I&=\left({115\,{\text{kg}}}\right)\left( {-3\,{\text{m/s}}\,\hat i}\right)\\&=345\,{\text{N.s}}\left({-\hat i} \right)\\\end{aligned}\)
Here, \(-I\) indicate that the impulse acts in opposite of east direction.
Thus, an impulse of \(\boxed{345\text{ N.s}}\) to the west is required to decrease the speed of the boat from \(13\text{ m/s}\) east to \(10\text{ m/s}\) 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.
