From a building three objects are dropped. First object with velocity u = 0 and time t1 and another object with velocity u and time t2 and another object with velocity u and time t3. Find the relationship between t1, t2 & t3

To analyze the scenario where three objects are dropped from a building, we need to consider the principles of free fall under the influence of gravity. The first object is dropped from rest, while the other two are dropped with an initial velocity. Let's break down the motion of each object and derive the relationship between their respective times of fall, t1, t2, and t3.

Understanding Free Fall

In free fall, the only force acting on the objects is gravity, which accelerates them downward at approximately 9.81 m/s² (assuming we are near the Earth's surface). The equations of motion for an object in free fall can be expressed as:

• For the first object (dropped from rest):

h = ut1 + (1/2)gt1²

Since the initial velocity (u) is 0, this simplifies to:

h = (1/2)gt1²

• For the second object (dropped with initial velocity u):

h = ut2 + (1/2)gt2²

• For the third object (also dropped with initial velocity u):

h = ut3 + (1/2)gt3²

Setting Up the Equations

Since all three objects are dropped from the same height (h), we can set their equations equal to each other:

(1/2)gt1² = ut2 + (1/2)gt2²

(1/2)gt1² = ut3 + (1/2)gt3²

Deriving Relationships

From the first equation, we can rearrange it to find a relationship between t1 and t2:

h = (1/2)gt1² implies t1² = (2h)/g

Substituting this into the second equation:

(1/2)g((2h)/g) = ut2 + (1/2)gt2²

This simplifies to:

h = ut2 + (1/2)gt2²

Now, we can express h in terms of t2:

h = ut2 + (1/2)gt2²

From this, we can derive a similar expression for t3:

h = ut3 + (1/2)gt3²

Final Relationships

Now we have two equations for h:

h = (1/2)gt1²

h = ut2 + (1/2)gt2²

h = ut3 + (1/2)gt3²

Setting these equal gives us:

(1/2)gt1² = ut2 + (1/2)gt2²

(1/2)gt1² = ut3 + (1/2)gt3²

From these equations, we can derive that:

t2 and t3 will be greater than t1 because they have an initial velocity (u) that contributes to their distance traveled. The exact relationship can be complex and depends on the values of u and the height h.

Conclusion

In summary, the relationship between t1, t2, and t3 can be expressed through the equations of motion for each object. The first object, dropped from rest, will take the least time to reach the ground, while the other two, having an initial velocity, will take longer. The specific relationship can be further explored by substituting known values for u and h, allowing for precise calculations of t2 and t3 in relation to t1.