Physics word problem?

Question

A hot air balloon is ascending straight up at a constant speed of 7.0 m/s. When the balloon is 12.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 30.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them has the same altitude at the same time. How far above ground level are these places?

Could you please put an explanation as well? I've been working on this for so long, but I am just no good at physics. Thank you.

Answer 1

let the ground level be the initial position of the pellet. The ballon is 12m above the ground, so 12m is the initial position of the ballon

the rate of the ball is the constant, we use:
Xf = vt + xi
Xf = 7t + 12

the rate of the pellet is an acceleration, we use:
Xf = .5at^2 + vt + xi
Xf = .5(-9.8)t^2 + 30t + 0

when the balloon and the pellet meet, they are in the same postion.

Xf(balloon) = Xf (pellet)

7t + 12 = .5(-9.8)t^2 + 30t + 0
-4.9t^2 + 23t - 12 = 0

use quadratic formula and you'll get t = 0.597898s and 4.09598s

so the meet at .597898s and at 4.09598s
plug in the times to find the distances from the ground.

Xf = 7(0.597898) + 12
Xf = 16.18m above the ground

Xf = 7(4.09598) + 12
Xf = 40.67m above the ground

Answer 2

At the time the bullet is fired, the equation for altitude of the balloon is

Hbn = Vbn*t + 12 (Hbn = balloon ht, Vbn = balloon velocity)

The bullet height is given by

Hbt = Vbt *t - 0.5*g*t^2 (Vbt = bullet initial velocity)

Equate these to find the time the heights are equal; you will have a quadratic equation, and there will be two times (one when bullet is rising, and one when bullet is falling):

Vbn*t + 12 = Vbt *t - 0.5*g*t^2

0.5*g*t^2 + (Vbn - Vbt)*t + 12 = 0

Solve for t1 and t2

Then plug those into either formula to get heights.

Answer 3

If we predict the motorcycles are traveling in the direction of one yet another, they'll meet in one hour. because of the fact the bee flies at 30 km/hr, it incredibly is going to return and forth 30 km in the only hour it takes the motorcycles to fulfill.