Parabolic Movement Questions

Question

Given The Equation for Parabolic Movement is h(t)=-16t²+v₀t+h₀. If an object is launched with an initial velocity of 64 ft/sec from a height of 80 feet. When will it land?

Would you also please include your work step by step so I can better understand how to do this problem! Thank you so much!

Answer 1

h(t) = -16t² + v₀t + h₀

h - height of the object above ground

t - time in the air

V₀ - initial velocity

h₀ - initial height

V₀ = 64 (ft/sec) ← given

h₀ = 80 (ft) ← given

When an object lands, the height of the object above the ground is zero.

Equaling the expression for "h" to "0" and solving for "t":

-16t² + 64t + 80 = 0 → divide by "-16"

t² - 4t - 5 = 0 → factoring

(t - 5)(t +1) = 0

t - 5 = 0, t = 5 (sec)

(t + 1) = 0, t = -1 (sec) → extraneous solution: time must be positive

Answer: 5 seconds in the air before hitting the ground.

Answer 2

Hello, Tae,

UPDATE: MY ANSWER IS INCORRECT. I USED ACCELERATION INSTEAD OF VELOCITY IN THE DERIVATION. PLEASE CHECK THE RESPONSE FROM YURI O.

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The equation tells us that the height at time t, h(t), is h(t)=-16t²+v₀t+h₀. where t is time.

The question wants to know the time it takes for the projectile to fall to Earth. This means the height is 0 (meters, feet, it doesn't matter).

The projectile is launched with an initial velocity of 64 feet/sec². We'll have to assume this is a vertical acceleration.

v₀ in the equation is the initial acceleration. We.re tempted to say 64 feet/sec², but this is a trap. We have to consider the force of gravity that is also acting on it at the time it was launch. So the net force is the sum of these two vectors. The value of 64 feet/sec² is added to gravity, at -32 feet/sec².Note that gravity is a negative value. The direction is down, so the vector of force is negative.

Add the two to get the net acceleration: (64 - 32) = 32 feet/sec².

h₀ is the initial height of 80 feet.

Use these v₀ and h₀ values in the equation and set it equal to zero (the height when it hits the ground.

h(t)=-C₀

0 =-16t²+32t+80

This is a parabala. It can be solved graphically by plotting the time as the x coordinate and the height as the y coordinate. One can also factor or use other direct mathematical means to find the solutions. One of two solutions is negative, which would mean going back in time. May be for a movie this would be allowed, but on Earth we can throw that one out. There is one positive solution, which by my graph is around 3.45 seconds.

If you use 3.95 seconds into the equation and solve for height, it comes to essentially zero.

I hope this helps,

Bob