A sled is initially given a shove up a frictionless 24.0° incline. It reaches a maximum vertical height 1.10 m higher than where it started. What was its initial speed?
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A sled is initially given a shove up a frictionless 19.0 ∘ incline. It reaches a maximum vertical height 1.35 m higher than where it started at the bottom.
To find the sled's initial speed, we can use the principle of conservation of mechanical energy. The total mechanical energy of the sled at the bottom of the incline (initial point) is equal to the total mechanical energy at the maximum height (final point) since there is no energy loss due to friction.
The total mechanical energy at any point can be calculated as the sum of the kinetic energy and the potential energy.
At the bottom of the incline, the sled has only kinetic energy, given by the equation:
KE = 0.5 * m * v^2
where KE is the kinetic energy, m is the mass of the sled, and v is the initial speed.
At the maximum height, the sled has only potential energy, given by the equation:
PE = m * g * h
where PE is the potential energy, m is the mass of the sled, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum vertical height.
According to the conservation of mechanical energy, the total mechanical energy at the bottom of the incline is equal to the total mechanical energy at the maximum height:
KE_initial = PE_final
0.5 * m * v^2 = m * g * h
We can cancel out the mass (m) from both sides of the equation:
0.5 * v^2 = g * h
Now, we can solve for the initial speed (v):
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Substituting the given values:
v = sqrt(2 * 9.8 * 1.10)
v ≈ 4.19 m/s
Therefore, the sled's initial speed was approximately 4.19 m/s.