The difference between speed and velocity
4 marks · NormalA boat travels at a constant speed of 5 m/s across a river that is 100 m wide. The current flows east at 3 m/s, and the boat points directly north (perpendicular to the current).
Explain the difference between the speed of the boat and its velocity as it crosses the river. Give your answer in terms of scalar and vector quantities, and describe the actual path of the boat across the river.
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Show answer
The speed of the boat is the magnitude of its movement, which is 5 m/s. Speed is a scalar quantity because it only has size. The velocity is a vector, so it considers both speed and direction. As the boat moves north but is pushed east by the current, its velocity is the combination of its northward speed (5 m/s) and the eastward current (3 m/s). The actual path of the boat is diagonal across the river, not straight north. The velocity vector points northeast, and its magnitude can be found using Pythagoras' theorem: in a direction northeast of north.
Must show: identifies speed as scalar, identifies velocity as vector, describes combining velocities/vectors, explains diagonal path
Also accept: Gives correct explanation about direction and magnitude for both speed and velocity
Common mistakes: Ignoring the direction of velocity; Describing speed as having a direction
Show mark scheme and worked solution
Marking points
B1 · 1 mark · States speed is a scalar quantity (has size only); gives 5 m/s as speed.
B2 · 1 mark · States velocity is a vector quantity (has both size and direction) and combines components.
B3 · 1 mark · Mentions that actual path is diagonal across the river due to both northward and eastward movement.
B4 · 1 mark · Describes the velocity as northeast with combined magnitude (e.g. 5.83 m/s or via Pythagoras).
Worked solution
The speed of the boat is 5 m/s, which tells us only how fast it moves, regardless of direction. Speed is a scalar quantity.
Velocity takes into account both the speed and the direction. Because the river pushes the boat east at 3 m/s and the boat moves north at 5 m/s, the true velocity is the vector sum of these two movements.
This means the boat moves diagonally across the river, so the velocity has both north and east components. The actual path is northeast as both the speed and direction are combined.