Final answer:
The initial potential energy at point A is 196 J. The velocity at point B is 7 m/s. The velocity at the top of the loop at point C is also 7 m/s. The spring will compress by approximately 0.33 m before coming to a stop.
Step-by-step explanation:
The initial potential energy of Snuffles and the sled at point A can be calculated using the formula for gravitational potential energy:
PEinitial = mgh
Where m is the mass (4 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height (5 m). Plugging in the values, we get:
PEinitial = (4 kg) * (9.8 m/s²) * (5 m) = 196 J
To find the velocity of Snuffles and the sled at point B, we can use the conservation of mechanical energy:
PEinitial + KEinitial = PEfinal + KEfinal
Since the potential energy at point B is zero (ground level), we can simplify the equation to:
KEinitial = KEfinal
Using the formula for kinetic energy:
KE = 0.5 * m * v²
Where m is the mass and v is the velocity, we can plug in the values and solve for v:
0.5 * 4 kg * vB² = 196 J
vB = √(196 J / (0.5 * 4 kg)) = 7 m/s
To find the velocity of Snuffles and the sled at the top of the loop at point C, we can use the principle of conservation of mechanical energy again:
PEinitial + KEinitial = PEfinal + KEfinal
At the top of the loop, the potential energy is:
PEfinal = mgh = (4 kg) * (9.8 m/s²) * (5 m) = 196 J
Using the same formula for kinetic energy, we can solve for vC:
KEinitial = 0.5 * m * vC²
0.5 * 4 kg * vC² = 196 J
vC = √(196 J / (0.5 * 4 kg)) = 7 m/s
To find how much Snuffles and the sled will compress the spring before coming to a stop, we can use Hooke's Law:
F = k * x
Where F is the force, k is the spring constant (120 N/m), and x is the displacement (compression of the spring). We need to find the displacement at which the force becomes zero, which is the maximum compression. Rearranging the equation, we get:
x = F / k
Plugging in the values, we get:
x = (4 kg * 9.8 m/s²) / 120 N/m ≈ 0.33 m