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A balloon starts ascending from the ground at a constant speed of 25m/s. After 5s a bullet is shot vertically upwards from the ground
What should be the minimum speed of the bullet so that it may reach the balloon?
If the bullet is shot at twice the speed calculated above find the height at which it passes the balloon

Rajalekshmy Ramaswamy , 10 Years ago
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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 vb.

Setting Up the Equations

When the bullet is shot, it will have an initial velocity vb and will be subject to gravitational acceleration, which we approximate as g=9.81m/s2. The height hb of the bullet after t seconds can be expressed as:

Height of the bullet: hb=vbt12gt2

At the same time, the height of the balloon after t seconds (which is 5 seconds plus the time t that the bullet is in the air) is:

Height of the balloon: hballoon=125+25t

Finding the Minimum Speed

For the bullet to reach the balloon, the heights must be equal:

Setting the two height equations equal gives us:

vbt12gt2=125+25t

Rearranging this equation, we can express it as:

vbt=125+25t+12gt2

Now, we need to find the minimum speed vb when the bullet is shot at the same time the balloon is at 125 m. To find vb, 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 tb.

Substituting tb into the equation, we can solve for vb when the bullet just reaches the balloon:

vb=125+25tb+12gtb2tb

Calculating Minimum Speed

To find the minimum speed, we can set tb=5 seconds (the time it takes for the bullet to reach the balloon). Plugging in the values:

vb=125+255+129.81525

Calculating this gives:

vb=125+125+122.6255=372.6255=74.525m/s

Height at Double the Speed

If the bullet is shot at twice this speed, then:

vbullet=274.525149.05m/s

Now, we can find the height at which the bullet passes the balloon. Using the same height equation for the bullet:

hb=149.05t129.81t2

Setting this equal to the height of the balloon:

149.05t129.81t2=125+25t

Rearranging gives:

149.05t25t129.81t2125=0

Which simplifies to:

124.05t4.905t2125=0

This is a quadratic equation in the form at2+bt+c=0. We can use the quadratic formula to solve for t:

t=b±b24ac2a

Substituting a=4.905, b=124.05, and c=125:

Calculating the discriminant and solving for t will give us the time at which the bullet passes the balloon. Once we have t, 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!

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