28. A boat leaves the dock at t = 0 and heads out into a lake with an acceleration of
2.0m/s2i^. A strong wind is pushing the boat, giving it an additional velocity of
2.0m/s iº+1.0m/s jº. (a) What is the velocity of the boat at t = 10 s? (b) What is the
position of the boat at t = 10s? Draw a sketch of the boat's trajectory and position at
t = 10 s, showing the x- and y-axes.

The addition of velocities must take into account that it is a vector magnitude, for which the modulus and direction of the resulting vector must be calculated.

The analytical method is one of the easiest ways to perform the addition of velocities, since the sum of the components is reduced to the algebraic sum and then we form the resulting vector.

In this case they indicate the velocities in the form of components of a Cartesian coordinate system, let's work each axis separately

a) speed

X axis

The final speed of the boat has two parts, one given by the acceleration of the boat and the other created by the thrust of the wind.

vₓ = v_{boat} + v_{x wind}

the wind speed is constant and the speed of the boat is searched with kinematics

v_{boat} = v₀ + aₓ t

Where v₀ is the initial velocity and aₓ the ecceleration.

When the boot leaves the dock its initial velocity is zero

v boat = 0 + aₓ t

v_{boat} = 2 10

v_{boat} = 20 m / s

The total speed of the boat is

vₓ = 20 + 2

vx = 22 m / s

Y axis

In this axis the boat has no initial velocity and the acceleration is zero

v_y = vybote + v_ {and wind}

v_y = 0 + 1.0

v_y = 1.0 m / s

The speed of the boat at 10 s is

v_{boat} = (22 i ^ + 1.0 j ^) m / s

b) The position of the boat

We work each axis separately

X axis

The position of the boat has two parts, one created by the acceleration of the boat and the other by what the wind pushes

x₁ = v₀ₓ t + ½ aₓ t²

x₁ = 0 + ½ 2 10²

x₁ = 100 m

the wind has a constant speed

x₂ = v_{x wind} t

x₂ = 2 10

x₂ = 20 m

Therefore the displacement of the boat on this axis is

x = 100 + 20

x = 120 m

Y axis

In this axis the boat has no initial speed or acceleration, so the only displacement is created by the wind

y = v_{y wind} t

y = 1.0 10

y = 10 m

We form the position vector of the boat

r = x i ^ + y j ^

r = (120 i^ + 10 j^) m

In conclusion with the relations of addition of vectors and kinematics we can find the results are;

a) v = (22 i^ + 1.0 j^) m / s

b) r = (120 i^ + 10 j^) m

Learn more about velocity addition and kinematics here:

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