A cat falls from the top of a 100 foot tree. the height of the cat, h(t), (in feet) can be modeled by the equation h(t) = -16t^2 +100. Find the velocity of the cat when it lands safely on the ground.

Question

I calculated the time when the cat reaches the ground, 2.5 seconds, but would the velocity be the 100 ft divided by this number? What am I missing?

Answer 1

If you computed 100 ft/ 2.5 s, then you have found the cat's average velocity in 2.5 s. The problem is asking for his instantaneous velocity at t = 2.5 s, which is the slope of the graph of h at t = 2.5 s. If you know about derivatives, the slope at t = 2.5 is dh/dt(t = 2.5 s), which is -32t at t = 2.5 s, i.e., -80 ft/s. If in case you have not yet learned about derivatives, you can use a graphing calculator to graph h and approximate the slope of the parabola at t = 2.5, which should be close to -80.

Answer 2

It would be helpful to know what you've learned in precalc at the point of this question.

In general, the velocity function v(t) will be the derivative of the height function h(t).

If you are not working with derivatives yet, did your teacher give you notes on how to find the velocity function from a height or distance function?

you have the right time t=2.5 s but now you would need to plug that into the velocity funtion.

Answer 3

h(t) = -16t^2 + 100

when the cat hits the ground, it's at time t

h(t) = 0 = -16t^2 +100 (h=0 means ground level, while ho=100 is the initial height)

16t^2 = 100

t^2 = 100/16

t = sqr(100/16)

= 10/4

= 2.5

take the derivative of h(t) to calculate velocity at time t

velocity = v(t) = h'(t) = -32t

velocity when the cat hits the ground

= v(2.5) = -32(2.5) = -80 meters per second

it's negative velocity to indicate the speed is downward

There's a Nobel Laureate in physics, Frank Wilczek, also 2022 Templeton Prize winner, who explains why the cat usually lands on its feet, but that still might not save its life or health, from 100 meters. The Nobel guy is into dark energy & matter, and new smaller subatomic particles. "Dark" might include black cats, things harder to see in the dark. "Black Cat Day" is a couple days before Halloween.

80 feet per second is the equivalent of 80/5,280 miles per 1/3,600 hour. 80 meters per seconds is over 3 times as fast. This problem sounds like serious animal abuse. Maybe a felony. PETA would not be amused. Velocity of 80 feet per second

= (80/5280)/(1/3600) = about 56 mph. 5280 feet per mile. 3600 seconds per hour. 3 times as fast is over 168 mph. A mosquito might stand a better chance on your windshield. Or more humane way to go.

While your idea of 100/2.5 = 40 is not correct, it's very interestingly exactly half the correct answer of 80 feet per second.

the higher the initial height, the higher the velocity when it hits the ground. So you're very much on the right track.

Let's say the initial height had been 10,000 meters

then

h(t) = -16t^2 +10,000

and

h(t) = 0 = -16t^2 +10000

t^2 = 10,000/16

t = 100/4 = 25 seconds

v(t) = -32t

v(25) = -32(25) = -800 m/sec

divide 10,000 by 25 = 400,

again exactly one half the actual speed

"velocity" usually has direction as well as magnitude, so it can be negative or positive

while "speed" usually is just the absolute value of velocity, always positive

Nonetheless, cat owners will not like this problem.

The -4.8t^2 and -9.6t terms are close approximations for the effect of gravity at sea level. It's not exact. Also get high enough up and the gravity force is reduced, but air resistance is also reduced given less air at higher altitudes, so it's almost a wash. For a while. If you want to quickly lose weight, just move to the mountains or Colorado and the scales will show you've lost a few pounds, due to less gravity. Go to space and you're weightless. But everyone generally ignores all these complications, for a problem like yours

If you did the 100/2.5 method, you get 40, so multiply it by -2 to get -80 m/sec. I'm curious what your instructor might say, if you did that. 2ho/t. The units match too, as feet divided by seconds = m/sec. which is a measure of velocity.

while some of these PETA or cat owner comments may seem totally irrelevant, actually many or most of these type problems are intended to be realistic or semi-realistic, so if the answers are too far off from what reality would dictate, you might want to recheck the calculations and arithmetic. It happens, that reality may help uncover a miscalculation somewhere.