Question

A motor vehicle moves such that the velocity is a linear function of time. The graph between acceleration-time and velocity-time, respectively will be

(a) Straight line, straight line

(b) Straight line , parabola

(c) Parabola, straight line

(d) Parabola, parabola

Answer 1

Since the velocity of the motor vehicle is a linear function of time, we can write it as:

v(t) = at + b

where "a" is the acceleration and "b" is the initial velocity.

Differentiating v(t) with respect to time, we get:

a(t) = dv/dt = a

which means that the acceleration is constant.

Therefore, the graph between acceleration-time will be a straight line, parallel to the time axis.

Integrating v(t) with respect to time, we get:

x(t) = (1/2)at^2 + bt + c

where "c" is the initial position.

Differentiating x(t) with respect to time, we get:

v(t) = dx/dt = at + b

which is the same as our original equation for velocity.

Therefore, the graph between velocity-time will be a straight line, sloping upwards, as the velocity increases with time.

Hence, the correct option is (a) straight line, straight line.

Answer 2

Since velocity is a linear function of time, its graph will be a straight line with a positive slope. Let's call this line "V-T" for convenience.

Now, we know that acceleration is the derivative of velocity with respect to time, which means that the graph of acceleration-time will be the slope of the velocity-time graph. Since the slope of a straight line is constant, the graph of acceleration-time will also be a straight line. Let's call this line "A-T".

Therefore, the correct answer is (a) Straight line, straight line.