Answer:
(a) 0.2618 J
(b) 0.1558 J
(c) 0 J
Explanation:
from Hook's Law,
The energy stored in a stretched spring = 1/2ke²
Ep = 1/2ke² ......................... Equation 1
Where k = spring constant, e = extension, E p = potential energy stored in the spring.
(a) When The spring is stretched to 4.11 cm,
Given: k = 310 N/m, e = 4.11 cm = 0.0411 m
Substituting these values into equation 1
Ep = 1/2(310)(0.0411)²
Ep = 155(0.0016892)
Ep =155×0.0016892
Ep = 0.2618 J.
(b) When the spring is stretched 3.17 cm
e = 3.17 cm = 0.0317 m.
Ep = 1/2(310)(0.0317)²
Ep = 155(0.0317)²
Ep = 155(0.0010049)
Ep = 0.155758 J
Ep ≈ 0.1558 J.
(c) When the spring is unstretched,
e = 0 m, k = 310 N/m
Ep = 1/2(310)(0)²
Ep = 0 J.
The equation for range is:
R = v₀²sin(2θ)/g
To find the maximum R, differentiate the equation and equate to zero. The solution is as follows:
dR/dθ = (v₀²/g)(sin 2θ)
dR/dθ = (v₀²/g)(cos 2θ)(2) = 0
cos 2θ = 0
2θ = cos⁻¹ 0 = 90
θ = 90/2
<em>θ = 45°</em>
Answer:
53.33 seconds
Explanation:
From the question;
- Power of the motor is 75 kW or 75000 W
- Depth or height is 150 m
- Volume of water is 400 m³
We are required to determine taken to raise the water from the given height.
We know that density of water is 1000 kg/m³
Therefore;
Mass of water = 400 m³ × 1000 kg/m³
= 4.0 × 10^5 kg
Thus, force required to raise the water;
= 4.0 × 10^5 kg × 10 N/kg
= 4.0 × 10^6 N
To determine the time;
we use the formula;
Time = work done ÷ power
= (4.0 × 10^6 N × 150 m) ÷ 75000 Joules/s
= 53.33 seconds
Therefore, time taken to raise the water is 53.33 seconds
Answer:
Explanation:
Consider that F (any function) <0 .
u(x,y) is a coontinuous function in the closed interval or region R.
Let us consider a point (p,q) that is inside the region and it is a maximum point.
Then it should be must
uxx (p,q) <0 where uxx means double differentiation
and uy(p,q) >0
Since ux(p,q) = 0 = uy(p,q) where ux and uy means single differentiation with respect to x and y respectively.
Say, Maximum limits of the region is T
therefore q<T
then uy (p,q) = 0 if q<T
if q = T then
point (p,q) = (p,T) will be on the boundary of R then we claim that
uy(p,q) >0
Similarly for the minimum also it will work
