A 10.3 kg block of ice slides without friction down a long track. The start of the track is 4.2 m higher than the end of the track, and the path traveled is 8.0 m. Find the speed of the block when it reaches the end of the track.
sinA = Y/r = 4.2 / 8 = 0.525,
A = 31.67deg.
Wb=10.3kg * 9.8N/kg = 100.9N @ 31.7deg = Weight of block.
Fp = 100.9sin31.7 = 53N = Force parallel to plane.
Ff = 0 = Force due to friction.
Fn=Fp - Ff = 53 - 0 = 53N = Net force.
a = Fn / m = 53 / 10.3 = 5.15m/s^2.
Vf^2 = Vo^2 + 2ad,
Vf^2 = 0 + 2 * 5.15 * 8 = 82.3,
Vf = 9.1m/s.
To find the speed of the block when it reaches the end of the track, we can use the principle of conservation of energy.
The potential energy of the block at the start of the track is given by the formula:
Potential energy = mass * acceleration due to gravity * height
Potential energy = 10.3 kg * 9.8 m/s^2 * 4.2 m
Next, we can calculate the final kinetic energy of the block at the end of the track using the formula for kinetic energy:
Kinetic energy = 0.5 * mass * velocity^2
Since there is no friction, all of the initial potential energy is converted into kinetic energy at the end of the track. Therefore, we can equate the potential energy and kinetic energy:
Potential energy = Kinetic energy
10.3 kg * 9.8 m/s^2 * 4.2 m = 0.5 * 10.3 kg * velocity^2
Simplifying the equation:
431.46 kg·m^2/s^2 = 5.15 kg * velocity^2
Divide both sides of the equation by 5.15 kg:
83.9 m^2/s^2 = velocity^2
Take the square root of both sides of the equation to solve for velocity:
velocity = √(83.9 m^2/s^2)
velocity = 9.16 m/s (rounded to two decimal places)
Therefore, the speed of the block when it reaches the end of the track is approximately 9.16 m/s.
To find the speed of the block when it reaches the end of the track, we can use the principle of conservation of energy. We know that at the start of the track, the block has potential energy (due to its height) and no kinetic energy (since it is at rest). At the end of the track, the block has no potential energy (since it has reached the bottom) and only kinetic energy (due to its motion).
The potential energy of an object of mass m at a height h is given by the equation: PE = mgh, where g is the acceleration due to gravity (9.8 m/s²).
The sum of potential energy (PE) and kinetic energy (KE) remains constant if there is no external work done by non-conservative forces, such as friction. This principle is known as conservation of mechanical energy, and it allows us to equate the initial potential energy to the final kinetic energy:
PE_initial + KE_initial = PE_final + KE_final
At the start of the track, the block has potential energy only:
PE_initial = mgh_initial = (10.3 kg)(9.8 m/s²)(4.2 m)
At the end of the track, the block has only kinetic energy:
KE_final = (1/2)mv²
Since the initial kinetic energy is zero, the equation becomes:
mgh_initial = (1/2)mv²
Now, we can solve for the velocity (v):
(1/2)mv² = mgh_initial
v² = 2gh_initial
v = √(2gh_initial)
Substituting the given values:
v = √(2 * 9.8 m/s² * 4.2 m)
v ≈ √(82.68) ≈ 9.1 m/s
Therefore, the speed of the block when it reaches the end of the track is approximately 9.1 m/s.