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 amount of energy the object loses, calculate the final kinetic energy - initial kinetic energy
First, we need to find the initial kinetic energy of the toy. The formula for kinetic energy is:
K = 1/2 * m * v^2
where K is the kinetic energy, m is the mass, and v is the velocity.
Given that the mass of the toy is 0.4 kg and the initial horizontal speed is 2.05 m/s, we can calculate the initial kinetic energy:
K_initial = 1/2 * 0.4 kg * (2.05 m/s)^2
= 0.418 J
Next, we need to find the final kinetic energy after the toy travels 0.5 meters beyond its launch point. Since the toy is subject to friction, it will lose some energy. The formula for work done by friction is:
W_friction = F * d * cosθ
where W_friction is the work done by friction, F is the force of friction, d is the distance, and θ is the angle between the force and displacement (in this case, θ = 180 degrees).
The force of friction can be calculated using the equation:
F_friction = μ * N
where F_friction is the force of friction, μ is the coefficient of kinetic friction, and N is the normal force. Since the toy is on a horizontal felt rug, the normal force equal to the toy's weight, which can be calculated using:
N = m * g
where m is the mass of the toy and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the coefficient of kinetic friction is 0.35, we can calculate the force of friction:
F_friction = 0.35 * (0.4 kg * 9.8 m/s^2)
= 1.372 N
The work done by friction is then:
W_friction = 1.372 N * 0.5 m * cos(180 degrees)
= -0.686 J
The negative sign indicates that the energy is lost.
Finally, we can calculate the final kinetic energy:
K_final = K_initial - W_friction
= 0.418 J - (-0.686 J)
= 1.104 J
Therefore, the velocity of the toy after traveling 0.5 meters beyond its launch point is given by:
K_final = 1/2 * m * v_final^2
1.104 J = 1/2 * 0.4 kg * v_final^2
v_final^2 = (2 * 1.104 J) / 0.4 kg
v_final^2 = 5.52 J/kg
Taking the square root of both sides, we find:
v_final = √(5.52 J/kg)
v_final ≈ 2.35 m/s
Therefore, the velocity of the toy after traveling 0.5 meters beyond its launch point is approximately 2.35 m/s.