To solve this problem, we need to use the principles of kinematics to describe the motion of both balls. Since both balls are thrown vertically upwards, we’ll focus on their velocities and displacements over time, while accounting for the acceleration due to gravity, which slows the balls down as they rise. The goal is to find when the two balls will meet, and at what height this happens.
Understanding the Problem
Let’s break down the details step by step. We have two balls with the following characteristics:
- Ball 1: Thrown with an initial velocity of 29.4 m/s.
- Ball 2: Thrown 3 seconds after Ball 1, with an initial velocity of 19.6 m/s.
Since the second ball is thrown 3 seconds later, the time when we are comparing the positions of the two balls is crucial. We will need to use the equations of motion for both balls and set their displacements equal to each other to find the collision time.
The Equations of Motion
The motion of each ball is governed by the equation:
h(t) = u t - ½ g t²
Where:
- h(t) is the height of the ball at time t,
- u is the initial velocity of the ball,
- g is the acceleration due to gravity (9.8 m/s²), and
- t is the time elapsed.
We can apply this formula for both balls.
For Ball 1
Since Ball 1 is thrown at time t = 0 with a velocity of 29.4 m/s, its height at time t is:
h₁(t) = 29.4 t - ½ (9.8) t²
For Ball 2
Ball 2 is thrown 3 seconds later, so its height at time t is given by:
h₂(t) = 19.6 (t - 3) - ½ (9.8) (t - 3)²
Note that t - 3 is the time elapsed for Ball 2, since it starts 3 seconds after Ball 1.
Finding the Time of Collision
To find when the two balls collide, we need to set their heights equal to each other, i.e., h₁(t) = h₂(t).
So, we equate the two expressions:
29.4 t - 4.9 t² = 19.6 (t - 3) - 4.9 (t - 3)²
Now, we solve this equation step by step:
- Expand both sides, simplify the terms, and solve for t.
- This results in a simple linear equation, which we can solve to find t = 5.25 seconds.
Determining the Height of the Collision
Now that we know the balls collide at t = 5.25 seconds, we can substitute this time back into either of the height equations to find the height of the collision. Let’s use the equation for Ball 1:
h₁(5.25) = 29.4 × 5.25 - ½ (9.8) (5.25)²
After calculating, we find:
h₁(5.25) = 18.6 meters
Conclusion
The two balls collide after 5.25 seconds at a height of 18.6 meters.
This type of problem is a good example of how you can use basic kinematic equations to analyze the motion of objects under the influence of gravity. The key idea is to carefully consider the timing of when each ball is thrown and solve for the time and height at which their paths intersect.
