# calculus and physics

posted by .

An explosion causes debris to rise vertically with an initial velocity of 75 ft/sec.
a. In how many seconds does it attain maximum height?
b. What is the maximum height?
c. What is the speed of the debris as it reaches 12 ft on the way back to the ground?
d. What is the velocity of the debris at 20 ft on the way up

• calculus and physics -

Easy way is to assume that total energy is constant (of course debris might have a lot of air drag but there is not much we can do about that.

mass m (will cancel everywhere in this problem)

(1/2) m v^2 + m g h = total energy
at start h = 0 and v = 75
so
(1/2)(75)^2 = total energy/m = 2813
in obsolete English units as here g = 32 ft/s^2
so at 12 ft
(1/2) v^2 = 32(12)
solve for v
and at 20 ft
(1/2) v^2 = 32 (20) solve for v

note that the speed is the same upwards and downwards at the same height, just the direction changes.

• Sorry, went too fast -

at top
(1/2) v^2 = 0
so
32 h = 2813

at 12 feet
(1/2)v^2 + 32 (12) = 2813

at 20 feet
(1/2) v^2 + 32 (20) = 2813

• calculus and physics -

a)
Height(t) = -16t^2 + 75t
Height ' (t) = -32t + 75 = 0 for a max of Height
-32t = -75
t = 75/32 seconds or appr 2.34 seconds

b) Height(75/32) = -16(75/32)^2 + 75(75/32) = 87.89 ft

c) when Height = 12
12 = -16t^2+ 75t
16t^2 - 75t + 12 = 0
t = 4.5216 sec (on the way down) or
t = .1659 seconds (on the way up)

velocity(t) = height ' (t)
= -32t + 75 , so we want it on the way down
velocity(4.5216) = -32(4.5216)+75 = -69.69 ft/sec

d)
same method as in c)
16t^2 - 75t + 20 = 0
t = 4.40 seconds (on the way down), or
t = .284 secpmds (on the way up

velocity(.284) = -32(.284)+75 = 65.92 ft/sec

## Similar Questions

1. ### ap calculus

The height s of a ball, in feet, thrown with an initial velocity of 112 feet per second from an initial height of 6 feet is given as a function of time t (in seconds) by s(t)=-16t^2+112t+20. 1. What is the maximum height of the ball?
2. ### calculus

If a ball is thrown vertically upward from the roof of 64 foot building with a velocity of 64 ft/sec, its height after t seconds is s(t)=64+64t–16t2 . What is the maximum height the ball reaches?
3. ### Calculus

If a ball is thrown vertically upward from the roof of 32 foot building with a velocity of 112 ft/sec, its height after t seconds is s(t)=32+112t–16t2 . What is the maximum height the ball reaches?
4. ### calculus and physics

An explosion causes debris to rise vertically with an initial velocity of 75 ft/sec. a. In how many seconds does it attain maximum height?
5. ### physics

A small ball is thrown vertically upwards with an initial velocity of 40m/s. Find:(1) its velocity after moving 3 seconds(2) the maximum heigh attained(3) the time to attain the maximum height(4)the total time taken for the ball to …
6. ### physics

A small ball is thrown vertically upwards with an initial velocity of 40m/s. Find:(1) its velocity after moving 3 seconds(2) the maximum heigh attained(3) the time to attain the maximum height(4)the total time taken for the ball to …
7. ### Calculus Help

If a ball is thrown vertically upward from the roof of 64 foot building with a velocity of 32 ft/sec, its height after t seconds is s(t)=64+32t–16t2. What is the maximum height the ball reaches "in ft"?
8. ### math

A body is thrown vertically upward from the ground with an initial velocity of 96 ft/sec. Find the maximum height attain by the body
9. ### Physics

A stone, 2kg is projected vertically at a velocity of 5m/s at an Angle of 60° Calculate I.Time taken to attain maximum height II.maximum height reached III.The range
10. ### Calculus

An explosion causes debris to rise vertically with an initial velocity of 160 feet per second. In how many seconds does it attain maximum height?

More Similar Questions