Question

If the distance traveled by a point in time $t$ is $s = 180t-16{t^2}$ , then the rate of change in velocity is
$(1){\text{ }}-16t{\text{ }}unit$
$(2)48{\text{ }}unit$
$(3) - 32{\text{ }}unit$
$(4)$ None of these

Answer 1

The given problem checks the concepts of derivatives and their uses. These can be easily solved if we take this derivative as a measure. Therefore we find the expressions for acceleration by the differentiation of the distance traveled by a point in time twice.

Complete step by step solution:
To be specific, when velocity is positive on an interval, we can find the total distance traveled by finding the area under the velocity curve on the given time interval. We may only be able to estimate this area, depending on the shape of the velocity curve. While taking the derivative of the position function we found the velocity function, and similarly, by taking the derivative of the velocity function we found the acceleration function. Making use of the integral calculus, we can work backward and compute the velocity function from the acceleration function, and also we can get the position function from the velocity function.
Let’s consider a particle with acceleration $a(t)$ is a known function of time. As the time derivative of the velocity function is called acceleration, the acceleration function varies linearly in time
Hence we get
$\dfrac{d}{{dt}}s(t) = v(t)$
And also,
$\dfrac{d}{{dt}}v(t) = a(t)$
By solving these two equations for the given values we get
$s = 180t-16{t^2}$
$\Rightarrow \dfrac{{ds}}{{dt}} = 180-32t$
$\Rightarrow \dfrac{{{d^2}s}}{{d{t^2}}} = -32$
Hence, change in velocity is $32{\text{ }}unit$
Hence, Option 3 is correct.

Note: Velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity can be given by meters per second or m/s, but many other units, like km/h, mi/h (also written as mph), and cm/s, are in common use. Instantaneous velocity v is the velocity at a specific instant or the average velocity for a very small interval. The theory that velocity is positive on a given interval assures that the movement of an object is always in a single direction, and therefore ensures that its change in position is equal to the distance it travels.