A) The objects have the same vertical position after 2 seconds
B) The objects have same vertical position at y = 30.4 m (but they do not collide since they have different x-position)
Explanation:
The motion of the first object along the vertical direction is a uniformly accelerated motion, so we can write its position at time t using the following equation:
where:
h = 50 m is the initial height
is the initial vertical velocity (the object is projected horizontally, so the vertical velocity is zero at the beginning)
is the acceleration of gravity
So, its vertical position can be rewritten as
The position of object 2 instead can be written as
where
is the initial vertical velocity, where
is the initial velocity
is the angle of projection
Substituting, we get:
The two objects collide when their vertical position is the same, so:
And solving for t, we find:
Note that this means that the two object at t = 2 s have the same vertical position: however, this is not true for the horizontal position.
B)
In order to find the point where they collide, we have to substitute the time of the collision that we found in part A into one of the expressions of the vertical position.
Substituting into the expression of object 2, we find:
We can verify that at the same time, the vertical position of object 1 is the same:
This means that the two objects have the same vertical position at 30.4 m.
However, in reality, the two objects do not collide. In fact, object 1 is moving in the horizontal direction with constant velocity
So its horizontal position at t = 2.0 s is
While object 2 is moving in the horizontal plane with velocity
So its horizontal position at t = 2.0 s is
So in reality, the two objects do not collide, if they start from the same x-position.
Learn more about projectile motion:
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