Physics A Level
Chapter 1: Kinematics 1.6 Combining velocities
Physics A Level
Chapter 1: Kinematics 1.6 Combining velocities
Velocity is a vector quantity and so two velocities can be combined by vector addition in the same way that we have seen for two or more displacements.
Imagine that you are attempting to swim across a river. You want to swim directly across to the opposite bank, but the current moves you sideways at the same time as you are swimming forwards. The outcome is that you will end up on the opposite bank, but downstream of your intended landing point. In effect,
you have two velocities:
- the velocity due to your swimming, which is directed straight across the river
- the velocity due to the current, which is directed downstream, at right angles to your swimming velocity.
These combine to give a resultant (or net) velocity, which will be diagonally downstream. In order to swim directly across the river, you would have to aim upstream. Then your resultant velocity could be directly across the river.
WORKED EXAMPLE
5) An aircraft is flying due north with a velocity of $200\,m\,{s^{ - 1}}$. A side wind of velocity $50\,m\,{s^{ - 1}}$ is blowing due east. What is the aircraft’s resultant velocity (give the magnitude and direction)?
Here, the two velocities are at ${90^ \circ }$. A sketch diagram and Pythagoras’s theorem are enough to solve the problem.
Step 1: Draw a sketch of the situation – this is shown in Figure $1.16a$.
Step 2: Now sketch a vector triangle. Remember that the second vector starts where the first one ends. This is shown in Figure $1.16b$.
Step 3: Join the start and end points to complete the triangle.
Step 4: Calculate the magnitude of the resultant vector v (the hypotenuse of the right-angled triangle).
$\begin{array}{l}
{v^2} = {200^2} + {50^2} = 40000 + 2500 = 42500\\
v = \sqrt {42500} \approx 206m\,{s^{ - 1}}
\end{array}$
Step 5: Calculate the angle $\theta $:
$\begin{array}{l}
\tan \theta = \frac{{50}}{{200}} = 0.25\\
\theta = {\tan ^{ - 1}}(0.25) \approx {14^ \circ }
\end{array}$
So the aircraft’s resultant velocity is $206\,m\,{s^{ - 1}}$ at ${14^ \circ }$ east of north, ${076^ \circ }$ or $N{76^ \circ }E$.
Questions
16) A swimmer can swim at $2.0\,m\,{s^{ - 1}}$ in still water. She aims to swim directly across a river that is flowing at $0.08\,m\,{s^{ - 1}}$. Calculate her resultant velocity. (You must give both the magnitude and the direction.)
17) A stone is thrown from a cliff and strikes the surface of the sea with a vertical velocity of $18\,m\,{s^{ - 1}}$ and a horizontal velocity v. The resultant of these two velocities is $25\,m\,{s^{ - 1}}$.
a: Draw a vector diagram showing the two velocities and the resultant.
b: Use your diagram to find the value of v.
c: Use your diagram to find the angle between the stone and the vertical as it strikes the water.