Two rocks launched up/collison (Kinematics)

[Moe777]

Homework Statement

This is a quiz question that was given in my AP physics C class. "A rock is launched upward with a speed V₀. With what velocity must a second rock be launched from the same initial position at time T later so that it hits the first rock when it has a velocity of -V₀/2."

Homework Equations

The only formulas we needed for this question were the basic velocity, displacement, and acceleration formulas (as functions of time) and their variations. (1D kinematics).

The Attempt at a Solution

1) I drew a diagram on an axis with a vertical line (y axis) and the origin being zero (the initial position of the balls).

2) I listed all of the knowns and unknowns given from the question. We know that:

• Rock₁ (first rock) has an initial velocity of V₀.
• Time= T
• Acceleration= -g= -9.81 m/s²
• When the second rock hits the first rock, the velocity of the first rock will be -V₀/2.

We don't know the velocity that is necessary for rock₂ (second rock), which allows the given condition ( When the second rock hits the first rock, the velocity of the first rock will be -V₀/2.) to happen. Basically, we don't know the initial velocity of rock₂.

I then assumed the conditions for the collision:
When the rocks collide, they will be the same distance from the ground, so d₁=d₂
and I also assumed the time of travel for both rocks at the collision, which I put as T=T, which is where I'm sure that I made a mistake.

Since this is from a collected quiz, I don't remember exactly what I did. However, I remember doing something similar to this:

(-V_0rock1 /2)T- (1/2)gT²=( V_0rock2)T- (1/2)gT²
I forgot the process, but my solution was something like V₀/2T, which is incorrect because the units come out to be m/s². I feel that I did something completely off track, and I really need help as to how I should approach these problems and how to think more effectively. I felt as though this question should have been easier for me, which kind of discouraged me after that quiz was collected. This isn't the first time we get a question like this, and help for the question and questions like these would be much appreciated. Thank you in advance.

 

[haruspex]

You seem to be confused about T. The given T is the interval between launch times. If the first rock is in flight for time t₁ before the collision, how long is the second rock in flight until then?

 

[mfb]

Time= T

That is not a good idea if more than one time is involved. Labels like t₁ and so on make it easier to see what happens when.

 

[Moe777]

Oh. I see what's going on. So I drew a graph just to help me visualize, where I assumed arbitrary times, where the first rock is thrown at 1s and the second is thrown at 4s. The time interval between them is 3s. This helped me confirm the equation that T=t₂- t₁, which means that t₂=t₁+T

 

[haruspex]

Oh. I see what's going on. So I drew a graph just to help me visualize, where I assumed arbitrary times, where the first rock is thrown at 1s and the second is thrown at 4s. The time interval between them is 3s. This helped me confirm the equation that T=t₂- t₁, which means that t₂=t₁+T

Maybe. It depends how you are defining t¹ and t₂. Are these times at which the rocks are thrown, or the times they spend in the air?

 

[Moe777]

I defined them as the times that the rocks were thrown.

 

[Moe777]

Ok. I still assumed that d1=d2.

V₀₁t₁+(1/2)gt₁²= V₀₂(T+t₁)+(1/2)g(T+t₁₎²

 

[mfb]

Now you gave the second object more time of flight than the first one.

Also, your use of t₁ doesn't match the definition above, as it seems to be the total flight time of the first object here.

 

[haruspex]

I defined them as the times that the rocks were thrown.

I'm still unclear what you mean. The times at which they were thrown or for which they were aloft?
If they are the times at which the rocks are thrown, you might as well choose t₁=0, making t₂=T. But then you need another variable to represent the time of the collision.
If they are the times the rocks spend in the air, which one should be the greater?

 

[Moe777]

Ok I think it's becoming clearer. I was going off of your first reply that T is the time interval between the launching of the rocks, and may have gotten confused from that. I now understand that T is the time in which they collide. Clearly, Rock1 is in the air longer because it was thrown first. Therefore, t₁ is greater than t₂. Therefore, the equation, T-t₁=t₂ must be true.

 

[haruspex]

Ok I think it's becoming clearer. I was going off of your first reply that T is the time interval between the launching of the rocks, and may have gotten confused from that. I now understand that T is the time in which they collide.

No, T is the time interval between launches. Reread the question.
What relationship does that give you between the times they spend in the air, t₁ and t₂?