Question

Why is the speed, in general, greater than the magnitude of the velocity?

Answer 1

First we will write the definition of the given parameters that is the speed and the velocity. Then using a diagram we can compare both magnitudes to find which one is greater. Here speed is a scalar quantity which does not depend on the direction of the body while velocity is a vector quantity which depends on the direction.

Complete step by step answer:
Speed is defined as the rate of change of position of a body in any direction as speed does not depend on the direction of motion of the body, so it is a scalar quantity. It is measured as the ratio of distance to the total time taken by the body to cover that distance.
Mathematically we can represent the speed as,
${\text{Speed = }}\dfrac{{{\text{Distance}}}}{{{\text{Time}}\,{\text{taken}}}}$
Velocity is defined as the rate of change of the body position with respect to the frame of reference and the time taken. It only depends on the initial and final position of the body; it is a path independent function. Basically velocity is called speeding in a specific direction so it is a vector quantity. It is measured as the ratio of displacement to the time taken by the body.
Mathematically we can represent the velocity as,
${\text{Velocity = }}\dfrac{{{\text{Displacement}}}}{{{\text{Time}}\,{\text{taken}}}}$

Now let us assume that a body first travels from point A to B then comes back to C.
Let the distance from A to C is x and B to C is y.
Now we can say that,
Total distance travelled = $x + 2y$
And total displacement = $x$
Now we can write speed as,
${\text{Speed = }}\dfrac{{{\text{Distance}}}}{{{\text{Time}}\,{\text{taken}}}} = \dfrac{{x + 2y}}{t}$
And velocity as,
${\text{Velocity = }}\dfrac{{{\text{Displacement}}}}{{{\text{Time}}\,{\text{taken}}}} = \dfrac{x}{t}$
On comparing both we will get,
${\text{Speed > Velocity}}$
Hence proved.

Note: We can also solve this problem theoretically, the average speed is not equal to the magnitude of the average velocity. This is because the motion involves a change in the direction and that is the reason that the path length is greater than the magnitude of the displacement and the velocity of a body depends on displacement while speed on distance or we can say path length. Hence the average speed is greater than the magnitude of the velocity.