Answer:
The speed is [tex]v = 4.425 m/s[/tex]
Explanation:
From the question we are told that
The spring constant is [tex]k = 75 \ N /m[/tex]
The mass of the foam dart is [tex]m = 5 g = \frac{5}{100} = 0.05 \ kg[/tex]
The compression distance is [tex]d = 10 cm = 0.1 m[/tex]
The height which the gun raised the dart is [tex]h = 5 m[/tex]
The change in height is [tex]\Delta h = 2 m[/tex]
The new height is [tex]h_2 = 5 -2 = 3 m[/tex]
Generally from the law of conservation of energy
[tex]E_s = KE[/tex]
Where [tex]E_s[/tex] is the energy stored in spring and it is mathematically represented as
[tex]E_s = \frac{1}{2} k d^2[/tex]
KE is the kinetic energy possessed by the dart when it is being shut and this is mathematically represented as
[tex]KE = \frac{1}{2} mv^2_r[/tex]
So
[tex]\frac{1}{2} k d^2 = \frac{1}{2} mv^2_r[/tex]
Substituting values
[tex]0.5 * 75 * 0.1 = 0.5 * 0.0005 * v^2_r[/tex]
=> [tex]v_r = \sqrt{\frac{0.5 * 75 * 0.1}{0.5 * 0.0005 } }[/tex]
[tex]v_r = 12.25 m/s[/tex]
When the dart is at the maximum height the
let it acceleration due air resistance be z
So by equation of motion
[tex]v^2 = u^2 - 2ah[/tex]
Where v is the velocity at maximum height which is equal to zero
and u is it initial velocity before reaching maximum height which we calculated as [tex]v_r = 12.25 m/s[/tex]
and a is the acceleration due to gravity + the acceleration due to air resistance
So
a = z+g
= 9.8 + z
=> [tex]v^2 = u^2 - 2(9.8 +z)h[/tex]
Substituting values
[tex]0 = 12.25^2 - 2(9.8 +z)h[/tex]
Making z the subject
[tex]z = \frac{ 12.25}{2 * 5} - 9.8[/tex]
[tex]z = \frac{ 12.25}{2 * 5} - 9.8[/tex]
[tex]z = 5 m/s[/tex]
When the dart is moving downward we can mathematically represent the motion as
[tex]v^2 = u^2 + 2ah[/tex]
Since the motion is downward and air resistance is upward we have that
a = g - z
and the the initial velocity u becomes the velocity at maximum height
i.e u = 0
And v is the velocity the dart has when it is moving downward
So
[tex]v^2 = 0 + 2 * (g -z )h[/tex]
Substituting values
[tex]v = \sqrt{0+ 2 (10 - 5) * 2}[/tex]
[tex]v = 4.425 m/s[/tex]
