A bullet is fired from a rifle with a horizontal velocity of 500 feet per second from a height of 16feet at the same time, another bullet falls out of the gun. without obstruction, which bullet will reach the ground first? how long will it take? At what velocity will it hit?
We will take the downward direction of motion as positive.
In the case of Bullet A
uah = Horizontal component of initial velocity ua
uav = Vertical component of initial velocity ua
It is convenient to resolve the velocity into horizontal and vertical components as these can be then dealt with independently of each other.
vah and vav are the horizontal and vertical components of final velocity va
In the case of bullet B there is no horizontal velocity component as the bullet is dropping vertically with no horizontal movement at all.
ub = initial velocity bullet B
vb = final velocity of bullet B
We choose to solve this problem by using the following equations of motion
v2 = u2 + 2 as
v = u + at
where v and u are the final and initial velocities of an object, a is the acceleration and s is distance traveled, t is time.
In this the horizontal component of velocity is of no consequence as it will not change at all, as there is no force acting on it ( in such problems we assume that there is no air resistance)
However, the vertical velocity will change due to the action of acceleration due to gravity.
We first use the equation vav2 = uav2 + 2 as
in this a = g = acceleration due to gravity = 32 ft/s2
uav = 0 m/s as the bullet is fired out in the horizontal direction
vav is what we want to determine
s = height = 16 ft
so vav2 = 0 + ( 2 x 32 x 16)= 322
hence vav = 32 ft/s
Now we use the other equation that is vav = uav + at
so t = (vav uav) / a
or t = (32-0)/32 = 1 s
Case of Bullet B
vb2 = ub2 + 2as
vb2 = 0 + (2 x 32 x 16) = 322
so vb = 32 ft/s
t = (vb - ub)/a
= (32-0)/32 = 1 s
So both bullets will hit the ground with the same vertical velocity and will take the same time to reach the ground
As bullet A also has a horizontal velocity of 500 m/s
So the actual velocity of bullet A will be
v = (vav2 + vah2)1/2 ( raising to the power of means taking the square root)
=(5002 + 322)1/2
= 501 ft/s