# A particle moves in a straight line in such a manner that s=1/2vt, s being the distance travelled in time t and v the velocity at the end of time t. prove that the acceleration is constant

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Motion in a single direction.

s = v t / 2

velocity = ds / dt

v = 1/2 [ v * 1 + t dv/dt ]

v = 1/2 [ v + a t ]

1/2 v = 1/2 at

v = a t

acceleration a = dv/dt = a * 1 + t * d a / dt

a = a + t * da/dt

=> da / dt = 0

=> a = dv/dt = acceleration = constant.

====================================

we can perhaps do the following way:

if acceleration is constant then, the equations of motion are:

v = u + at

s = u t + 1/2 a t²

v² - u² = 2 a s

in our context, s = 1/2 v t

v = 2 s / t, differentiate wrt time t:

a = dv/dt

= 2 [t * ds/dt - s * 1 ] / t²

= 2 [ v t - s ] / t²

= 2 [ v t - 1/2 vt ] / t²

= 2 * vt/2 /t²

= v / t

da/dt = [ t * dv/dt - v * 1 ] /t²

= [ t * a - v ] / t²

= 0 as v = a t

Hence, Acceleration is constant. as its derivative is zero.

s = v t / 2

velocity = ds / dt

v = 1/2 [ v * 1 + t dv/dt ]

v = 1/2 [ v + a t ]

1/2 v = 1/2 at

v = a t

acceleration a = dv/dt = a * 1 + t * d a / dt

a = a + t * da/dt

=> da / dt = 0

=> a = dv/dt = acceleration = constant.

====================================

we can perhaps do the following way:

if acceleration is constant then, the equations of motion are:

v = u + at

s = u t + 1/2 a t²

v² - u² = 2 a s

in our context, s = 1/2 v t

v = 2 s / t, differentiate wrt time t:

a = dv/dt

= 2 [t * ds/dt - s * 1 ] / t²

= 2 [ v t - s ] / t²

= 2 [ v t - 1/2 vt ] / t²

= 2 * vt/2 /t²

= v / t

da/dt = [ t * dv/dt - v * 1 ] /t²

= [ t * a - v ] / t²

= 0 as v = a t

Hence, Acceleration is constant. as its derivative is zero.