A 1.80-kg block slides down a frictionless ramp.The top of the ramp is h1 = 1.21 m above the ground; the bottom of the ramp is h2 = 0.256 m above the ground. The block leaves the ramp moving horizontally, and lands a horizontal distance $d$ away. Calculate the distance d.
(1/2) m u^2 = m g (Hup - Hdown)
so
u = sqrt(2*9.81*.954)
u is horizontal speed, does not change
How long in the air? Well, how long to fall .256 m ?
.256 = 4.9 t^2
solve for t, time in the air
distance horizontal = d = u t
To calculate the distance d, we can use the principle of conservation of energy. The total mechanical energy of the system is conserved, which means that the initial potential energy (mgh1) is equal to the final kinetic energy (0.5mv^2) plus the final potential energy (mgh2).
Step 1: Calculate the initial potential energy
Given that the mass of the block is 1.80 kg and the height of the ramp h1 is 1.21 m, we can calculate the initial potential energy:
Initial Potential Energy (P.E.i) = m * g * h1
= 1.80 kg * 9.8 m/s^2 * 1.21 m
Step 2: Calculate the final potential energy
Given that the height h2 is 0.256 m, we can calculate the final potential energy:
Final Potential Energy (P.E.f) = m * g * h2
= 1.80 kg * 9.8 m/s^2 * 0.256 m
Step 3: Calculate the change in potential energy
The change in potential energy is given by the difference between the initial and final potential energy:
Change in Potential Energy (Delta P.E) = P.E.f - P.E.i
Step 4: Calculate the final kinetic energy
Since the block leaves the ramp moving horizontally, its final velocity (v) will be constant. We can use the fact that kinetic energy (K.E) is given by 0.5 * m * v^2.
Step 5: Equate the change in potential energy and the final kinetic energy
Since energy is conserved, we can set the change in potential energy equal to the final kinetic energy:
Delta P.E = K.E.f
Step 6: Solve for the final velocity
Solve the equation from step 5 for the final velocity:
v^2 = 2 * Delta P.E / m
Step 7: Calculate the distance traveled
The distance traveled by the block can be calculated using the equation of motion for horizontal motion:
d = v * t
Since the block leaves the ramp moving horizontally, it will travel the same horizontal distance as the distance it falls vertically. So the distance traveled will be equal to h2.
Substitute the value of v (from step 6) into the equation for d to calculate the distance traveled by the block.