Solving for Multiple Objects

Often situations arise where we have more than one object moving. We have learned to apply kinematic equations to one object, but how can we use them for more than one object? When solving for multiple objects, we must find relationships between objects and create equations based on those relationships. You must spend time thinking of the relationships between the variables. Usually the relationships can be found with the time of motion for the two objects or the displacement of the two objects. For example, if one object has a 5 second head start, then we could write that relationship as LaTeX: t_1=t_2+5 . These relationships are used to generate equations. Problem solving involves applying an appropriate kinematic equation to each object and then creating equations from the relationships between the two. Remember that for every variable present we must have an equation. Four variables means that we must write four different equations. Please watch the following presentation on solving for multiple objects. Please watch the presentation on the Solving for Multiple Objects.

Solving for Multiple Objects Practice

Solving for Multiple Objects Self-Assessment

Now it is your turn. Complete the self-check questions to verify that you understand how to solve problems dealing with two objects moving at the same time.

A man and a women are out jogging. The man is 25 m east of a park bench, and is running west at a constant 4 m/s. The woman is 15 m west of the park bench and has just gotten done tying her shoe. She accelerates at 0.5 m/s² east toward the park bench. In relation to the bench, at what point will they pass each other? Let's walk through this problem step by step. Try to complete each part before rolling over the solution.
1. Both are in motion for the same period of time, so SOLUTION
2. The displacement will be the sum of their distances from the bench. SOLUTION
3. The equation for the man who is at constant velocity must be SOLUTION
4. The equation for the women, since she starts with an initial velocity of zero will be SOLUTION
5. Because the times are the same, we can replace the t_M and t_W with t. SOLUTION
6. We can find the displacement of either person. Combine the first and second equations. SOLUTION
7. Now combine with the remaining equation. SOLUTION
8. And solve SOLUTION
9. Using the quadratic equation we get SOLUTION

The question asked how far from the park bench they passed. Since she was 15 m from the park bench, they would pass each other 2.9 m west of the park bench.

A man is racing his son to a tree. He gives his son a 5 second head start and then accelerates at 2m/s². His son runs at a constant 3 m/s. The two reach the tree at the same time. For how long a time did the son run? Let's walk through this problem step by step. Try to complete each part before rolling over the solution.
1. The distance the two ran is the same. SOLUTION
2. The son gets to run for 5s longer than his father. SOLUTION
3. The son runs at a constant velocity. We are using x for displacement from this point forward because they both have the same displacement. SOLUTION
4. The man runs starting from rest and accelerating. SOLUTION
5. We want to solve for the times, so combine the last two equations. SOLUTION
6. Manipulate the time equation to solve for the man's time, the substitute into the equation above. That will leave us with only the time of the son in the equation. SOLUTION
7. Solve SOLUTION
8. Use the quadratic equation to solve. The first solution would not work because it is before the father started. SOLUTION

Solving for Multiple Objects Practice Problems

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