Question

A particle moves along x-axis in such a way that its x-coordinate varies with time according to the equation x= 4-2t+t^2. the speed of the particle will vary with time as?

Answer 1

The position of the object is given as a function of time:

$x = 4 - 2t + t^2$

Speed is the rate of change of position with respect to time. So, we can find speed by differentiating position with respect to time.

$x =4-2t+t^2 \\ \\ \\ v = \frac{dx}{dt} \\ \\ \\ \implies v = \frac{d}{dx} (4-2t+t^2) \\ \\ \\ \implies v = 0 - 2 + 2t \\ \\ \\ \implies \boxed{v = 2t-2}$


Thus, speed varies with time according to the function v = 2t-2

Answer 2

The correct answer to the question is 2t-2 and speed varies linearly with time.

EXPLANATION:

As per the question, the particle is moving along X-axis.

The distance is given as a function of time.

The distance (position) is given as $x\ =\ 4-2t+t^2$

First of all, we have to calculate the speed of the particle.

The speed of a particle is defined as the rate of change of position.

Hence, the instantaneous speed can be calculated by simply differentiating position with respect to time.

Mathematically speed $v\ =\ \frac{d}{dt}(x)$

                                           $=\ \frac{d}{dt}(4-2t+t^2)$

                                           $=\ \frac{d}{dt}(2)-\frac{d}{dt}(2t)+\frac{d}{dt}(t^2)$

                                          $=\ 0-2+2t$        [∵ $\frac{d}{dt}(c)=0$ ]

Here, c stands for a constant.

Hence, the speed as a function of time is given as v = 2t-2.

From above, it is obvious that the relation is similar to y = mx+c.

Hence, speed varies linearly with time.