Baseball of 0.15 kg is thrown upward at an initial velocity of 35 m/s . tow seconds later a 20 g bullet is fired at 250 m/s into the rising baseball . How high will the combined bullet and baseball rise.
To find the maximum height reached by the combined baseball and bullet, we need to determine the time it takes for the baseball to reach its highest point.
Step 1: Find the time it takes for the baseball to reach its highest point.
Using the kinematic equation: vf = vi + at, where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time, we can rearrange the equation to solve for time (t):
0 = 35 m/s - 9.8 m/s^2 * t
Simplifying this equation gives us:
9.8 m/s^2 * t = 35 m/s
t = 35 m/s / 9.8 m/s^2
t ≈ 3.57 s (rounded to two decimal places)
Step 2: Find the height reached by the baseball when it's at its highest point.
Using the kinematic equation: d = vit + 0.5at^2, where d is the displacement, vi is the initial velocity, a is the acceleration, and t is the time, we can substitute the values:
d = (35 m/s) * 3.57 s + 0.5 * (-9.8 m/s^2) * (3.57 s)^2
Simplifying this equation gives us:
d ≈ 62.04 m (rounded to two decimal places)
Step 3: Find the time it takes for the bullet to reach the same height as the baseball.
Since we know the initial velocity of the bullet, we can use the kinematic equation to find the time it takes for the bullet to reach the same height as the baseball.
Using the equation: d = vi * t + 0.5 * a * t^2, where d is the displacement, vi is the initial velocity, a is the acceleration, and t is the time, we can rearrange the equation to solve for time (t):
(62.04 m) = (250 m/s) * t + 0.5 * (-9.8 m/s^2) * t^2
Simplifying this equation gives us a quadratic equation:
4.9 t^2 - 250 t + 62.04 = 0
Using the quadratic formula, t can be solved as:
t = [-(-250) ± √((-250)^2 - 4 * 4.9 * 62.04)] / (2 * 4.9)
Simplifying this equation gives us two solutions for t:
t ≈ 2.36 s (rounded to two decimal places) or t ≈ 10.42 s (rounded to two decimal places)
Step 4: Find the maximum height reached by the combined bullet and baseball.
Since the bullet and baseball reach their highest points at different times, we need to consider the scenario where they reach their highest points simultaneously.
The bullet takes less time to reach the maximum height, so we will use the time when the bullet reaches its highest point, which is approximately 2.36 seconds.
Using the kinematic equation: d = vit + 0.5 * a * t^2, we substitute the values:
d = (250 m/s) * 2.36 s + 0.5 * (-9.8 m/s^2) * (2.36 s)^2
Simplifying this equation gives us:
d ≈ 284.5 m
Therefore, the combined bullet and baseball will rise to a maximum height of approximately 284.5 meters.
To determine the height at which the combined bullet and baseball will rise, we need to analyze the motion of each object separately and then find the point of intersection.
Let's begin with the baseball:
1. Find the time it takes for the baseball to reach its maximum height:
- The initial velocity of the baseball is 35 m/s in the upward direction.
- The acceleration due to gravity is approximately 9.8 m/s^2, acting in the downward direction.
- Using the formula v = u + at (where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time), we can calculate the time it takes for the baseball to reach its maximum height.
- In this case, v will be 0 m/s at the maximum height since the ball momentarily stops.
- Therefore, the equation becomes 0 = 35 - 9.8t.
- Rearranging the equation, we get t = 35/9.8.
- Calculating this, t ≈ 3.57 seconds.
2. Find the maximum height reached by the baseball:
- We can use the kinematic equation s = ut + (1/2)at^2 (where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time) to find the maximum height.
- Plugging in the values, s = 35(3.57) + (1/2)(-9.8)(3.57)^2.
- Evaluating this, we get s ≈ 62.64 meters.
Now let's analyze the motion of the bullet:
1. Since the bullet is fired two seconds after the baseball, we need to consider the additional time for the bullet.
- Therefore, the total time for the bullet will be the time calculated for the baseball (3.57 seconds) plus the 2 seconds of delay.
- Thus, the total time for the bullet is 3.57 + 2 = 5.57 seconds.
2. Find the maximum height reached by the bullet:
- Using the same kinematic equation as before, s = ut + (1/2)at^2, we can calculate the displacement.
- The initial velocity of the bullet is 250 m/s in the upward direction.
- The acceleration due to gravity remains the same, approximately 9.8 m/s^2, acting downward.
- Plugging in the values, s = 250(5.57) + (1/2)(-9.8)(5.57)^2.
- By calculating this, we find that s ≈ 666.89 meters.
Finally, to determine the combined height, we add the heights of the baseball and the bullet together:
Combined height ≈ 62.64 m + 666.89 m ≈ 729.53 meters.
Therefore, the combined bullet and baseball will rise to approximately 729.53 meters.