Askiitians Tutor Team
Last Activity: 14 Days ago
To determine the minimum speed of the bullet needed to reach the balloon, we first need to understand the motion of both the balloon and the bullet. The balloon ascends at a constant speed, while the bullet will have an initial velocity that we need to calculate. Let's break this down step by step.
Understanding the Balloon's Motion
The balloon rises at a constant speed of 25 m/s. After 5 seconds, we can calculate how high the balloon has ascended:
- Height of the balloon after 5 seconds = speed × time
- Height = 25 m/s × 5 s = 125 m
So, after 5 seconds, the balloon is at a height of 125 meters.
Bullet's Motion
Now, let's consider the bullet. It is shot vertically upwards after the balloon has already been ascending for 5 seconds. To reach the balloon, the bullet must travel the same height as the balloon at the same time. We need to find the minimum speed of the bullet, which we will denote as .
Setting Up the Equations
When the bullet is shot, it will have an initial velocity and will be subject to gravitational acceleration, which we approximate as . The height of the bullet after seconds can be expressed as:
Height of the bullet:
At the same time, the height of the balloon after seconds (which is 5 seconds plus the time that the bullet is in the air) is:
Height of the balloon:
Finding the Minimum Speed
For the bullet to reach the balloon, the heights must be equal:
Setting the two height equations equal gives us:
Rearranging this equation, we can express it as:
Now, we need to find the minimum speed when the bullet is shot at the same time the balloon is at 125 m. To find , we can assume that the bullet reaches the balloon at the same time it takes for the balloon to rise further. Let's denote the time taken for the bullet to reach the balloon as .
Substituting into the equation, we can solve for when the bullet just reaches the balloon:
Calculating Minimum Speed
To find the minimum speed, we can set seconds (the time it takes for the bullet to reach the balloon). Plugging in the values:
Calculating this gives:
Height at Double the Speed
If the bullet is shot at twice this speed, then:
Now, we can find the height at which the bullet passes the balloon. Using the same height equation for the bullet:
Setting this equal to the height of the balloon:
Rearranging gives:
Which simplifies to:
This is a quadratic equation in the form . We can use the quadratic formula to solve for :
Substituting , , and :
Calculating the discriminant and solving for will give us the time at which the bullet passes the balloon. Once we have , we can substitute it back into either height equation to find the height at which the bullet passes the balloon.
This approach allows us to understand the dynamics of both objects and calculate the required values systematically. If you have any further questions or need clarification on any step, feel free to ask!