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 determine the velocity of the toy after traveling 0.5 meters, we need to analyze the forces acting on it and use the concept of energy conservation.
1. Calculate the work done by the kinetic friction force:
The equation for the work done by friction is given by:
Work = Force x Distance x cos(180°) = μk * N * d * (-1),
where μk is the coefficient of kinetic friction, N is the normal force (equal to the weight of the toy), and d is the distance traveled.
The normal force N is given by:
N = m * g,
where m is the mass of the toy and g is the acceleration due to gravity.
Substituting the known values:
N = (0.4 kg) * (9.8 m/s^2) = 3.92 N
Work = (0.35) * (3.92 N) * (0.5 m) * (-1) = -0.686 J
2. Calculate the work done by the spring:
The work done by the spring can be calculated using the equation:
Work = (1/2) * k * (x2^2 - x1^2),
where k is the spring constant (200 N/m), x2 is the final position of the spring (0.1 m), and x1 is the initial position of the spring (0.04 m).
Substituting the known values:
Work = (1/2) * (200 N/m) * ((0.1 m)^2 - (0.04 m)^2) = 0.84 J
3. Determine the total work done on the toy:
The total work done on the toy is the sum of the work done by the friction force and the work done by the spring:
Total Work = Work (friction) + Work (spring)
Total Work = -0.686 J + 0.84 J = 0.154 J
4. Use the principle of conservation of energy:
According to the principle of conservation of energy, the total work done on an object is equal to the change in its kinetic energy:
Total Work = ΔKE = (1/2) * m * v^2,
where ΔKE is the change in kinetic energy and v is the velocity of the toy.
Rearranging the equation:
v^2 = (2 * Total Work) / m
v^2 = (2 * 0.154 J) / 0.4 kg
v^2 = 0.77 J/kg
v = sqrt(0.77 J/kg)
v ≈ 0.88 m/s
Therefore, the velocity of the toy after traveling 0.5 meters beyond its launch point is approximately 0.88 m/s.