Question

An iceboat sails across the surface of a frozen lake with constant acceleration produced by the wind. At a certain instant the boat’s velocity is$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{30}\hat{\mathbf{i}}\mathbf{-}\mathbf{8}\mathbf{.}\mathbf{42}\hat{\mathbf{j}}\mathbf{)}\mathbf{m}\mathbf{/}\mathbf{s}$. Three seconds later, because of a wind shift, the boat is instantaneously at rest. What is its average acceleration for this $\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{s}$interval?

Short answer

The average acceleration of an iceboat is$\left(-2.1\hat{i}+2.8\hat{j}\right)m/s^2$

Step-by-step solution

The given data

1) The boat’s velocity at an instant,$V_i=\left(6.30\hat{i}-8.42\hat{j}\right)m/s$

2) The time interval,$t=3s$

Understanding the concept of the average acceleration

The rate at which the velocity changes is referred to as average acceleration. Using the equation of average acceleration and the equation for the magnitude of velocity from its components, we can calculate the average acceleration of an iceboat.

Formula:

The average acceleration of a body in motion,$a_{avg}=\frac{V_f{V}_i}{t}$ …(i)

Calculation of the average acceleration of the particle

Using the given data in equation (i), we get the average acceleration of the particle as follows:

$$\begin{gathered} a_{avg}=\frac{0-\left(6.30\hat{i}-8.42\hat{j}\right)m/s}{3s} \\ =\left(-2.1\hat{i}+2.8\hat{j}\right)m/s^2 \end{gathered}$$

Hence, the value of the average acceleration is $\left(-2.1\hat{i}+2.8\hat{j}\right)m/s^2$