Question

Show that for a projectile d

2

(v2

) / dt2

= 2g2

Answer 1

We have to prove that,
d² (v²)  / dt² = 2 g²
Here we consider the vertical component of the velocity.
So, we take, v = v₍y₎
Hence,
d² (v²)  / dt² = d² (v₍y₎²)  / dt²
d² (v²)  / dt² = d / dt [ 2 v₍y₎ ( d v₍y₎ / dt ) ]   ........... (1)
d v₍y₎ / dt = Acceleration in vertical direction = - g  ............. (2)
Put in equation (1), we get:
d² (v²)  / dt² =  d / dt ( - 2 g v₍y₎ )
d² (v²)  / dt² = -2 g  ( d v₍y₎ / dt )
d² (v²)  / dt² = - 2 g ( - g )  ,   Again using (2)
d² (v²)  / dt² = 2 g²
Hence proved.
Hopefully it is helpful to you.
Thanks.