If the 0.4 kg spring were used to launch the toy horizontally across a felt rug, with a coefficient of kinetic friction being 0.35, what would be the velocity of the toy after traveling 0.5 meter beyond its launch point?
The max height of the pop up is 0.214 m.
the initial horizontal speed of the pop up toy when it’s released would be 2.05 m/s.
The spring constant is 200 N/m
The energy stored in the spring of it’s stretched from 0.04 to 0.1 m is 0.84 J
To answer this question first take the amount of energy the spring starts with and subtract the energy the object loses over 0.5 m
To find the velocity of the toy after traveling 0.5 meters beyond its launch point, we can use the principle of conservation of energy.
The initial energy stored in the spring is given as 0.84 J. Subtracting the energy the object loses over 0.5 meters will give us the remaining energy available for the toy's motion.
The energy lost by the toy over a distance, due to friction, can be calculated using the formula:
Energy lost = force of friction × distance
The force of friction can be found using the formula:
Force of friction = coefficient of kinetic friction × normal force
The normal force is equal to the weight of the toy, which can be calculated using the formula:
Normal force = mass × acceleration due to gravity
Given:
Mass of the toy (m) = 0.4 kg
Coefficient of kinetic friction (μ) = 0.35
Distance traveled beyond launch point (d) = 0.5 m
Spring constant (k) = 200 N/m
Initial horizontal speed of the toy (v0x) = 2.05 m/s
First, let's calculate the force of friction:
Normal force = mass × acceleration due to gravity
Normal force = 0.4 kg × 9.8 m/s^2
Normal force = 3.92 N
Force of friction = coefficient of kinetic friction × normal force
Force of friction = 0.35 × 3.92 N
Force of friction = 1.37 N
Next, let's calculate the energy lost by the toy over 0.5 meters:
Energy lost = force of friction × distance
Energy lost = 1.37 N × 0.5 m
Energy lost = 0.685 J
Now, let's find the remaining energy available for the toy's motion:
Remaining energy = Initial energy stored in the spring - Energy lost
Remaining energy = 0.84 J - 0.685 J
Remaining energy = 0.155 J
Finally, let's use the remaining energy to calculate the velocity of the toy after traveling 0.5 meters:
Remaining energy = (1/2) × mass × velocity^2
0.155 J = (1/2) × 0.4 kg × velocity^2
0.155 J = 0.2 kg × velocity^2
Solving for velocity:
velocity^2 = 0.155 J / 0.2 kg
velocity^2 = 0.775 m^2/s^2
Taking the square root of both sides:
velocity ≈ 0.88 m/s
Therefore, the velocity of the toy after traveling 0.5 meter beyond its launch point is approximately 0.88 m/s.