Question

An object is moving with acceleration but with uniform speed. Is it possible or not?Give examples.

Answer 1

Hint:To solve this problem learn the concept of acceleration and velocity properly. The acceleration of any object is the rate of change of its velocity with time.The rate of change of velocity with respect to time is called the acceleration.

Formula used:
The expression of acceleration is given by,
\[\vec a = \dfrac{{\vec v}}{t}\]
where \[\vec a\] is the acceleration of the body, \[\vec v\] is the velocity of the body and \[t\] is the time.
The expression of velocity is given by,
\[\vec v = \dfrac{{\vec s}}{t}\]
where \[\vec s\] is the displacement of the body, \[\vec v\] is the velocity of the body and \[t\] is the time.

Complete step by step answer:
Acceleration is nothing but at which rate velocity changes of a body or a particle. The mathematical expression of acceleration is given by, \[\vec a = \dfrac{{\vec v}}{t}\] where \[\vec a\] is the acceleration of the body, \[\vec v\] is the velocity of the body and \[t\] is the time. The instantaneous acceleration of a body is given by, \[\vec a = \dfrac{{d\vec v}}{{dt}}\] where, \[d\vec v\]is the change in velocity when the change in time is tends to zero, \[\mathop {dt = \lim }\limits_{\Delta t \to 0} \Delta t\].

We also know, the velocity of a body is the rate of change of its displacement per unit time. The expression of velocity is given by, \[\vec v = \dfrac{{\vec s}}{t}\] where \[\vec s\] is the displacement of the body. Velocity and acceleration both are vector quantities and depend on the direction of motion of the body but speed is a scalar quantity and it is the rate of change of distance with time.

Now, we have asked if a body moves with uniform speed can it have acceleration. Now, if a body moves in a linear path with uniform speed or that means with uniform velocity the body cannot have any acceleration related to it.
\[\vec v = \vec k\] [\[k\]is a constant vector]
\[\Rightarrow \vec a = \dfrac{{d\vec v}}{{dt}} = 0\]

But if the body moves in a circular path with uniform speed the instantaneous velocity of the body is not constant or uniform since the direction of motion changes with time hence the instantaneous acceleration of the body cannot be zero. Here, \[\vec v = k\hat r\] where, \[\hat r\] is the unit position vector.
\[\vec a = \dfrac{{d\vec v}}{{dt}} \ne 0\] [since, \[\dfrac{{d\hat r}}{{dt}} \ne 0\]]
So, if a body orbits in a circular path the instantaneous acceleration of the body is not zero.
So, in such cases where the magnitude of velocity is constant but direction of it changes with time then the body moving with acceleration can have uniform speed.

Note:If the body moves in a circular path constantly the instantaneous acceleration of the body cannot be zero but the net acceleration of the body over a full circle is zero since the net displacement of the body is zero. When the body moves in a circular path the magnitude of instantaneous velocity and the speed is the same for both of them.