The angle between the ramp and the horizontal is approximately 93.3°.
We will solve this problem using the following steps:
1. Identify the relevant equations:
Two equations are relevant to this problem:
Conservation of energy: This principle states that the total mechanical energy of a closed system is constant. In this case, the total mechanical energy of the ice block consists of its initial potential energy (at the top of the ramp) and its final kinetic energy (at the bottom of the ramp).
Kinematic equation: Since the ramp is frictionless, the only force acting on the block is gravity. Therefore, the block undergoes a constant acceleration down the ramp. We can use the following kinematic equation to relate the block's initial and final velocities, acceleration, and distance:
v^2 = u^2 + 2as
where:
v is the final velocity (2.96 m/s)
u is the initial velocity (0 m/s)
a is the acceleration
s is the distance (1.60 m)
2. Apply the conservation of energy:
Initially, the ice block has only potential energy:
PE = mgh
where:
m is the mass of the ice block (8.50 kg)
g is the acceleration due to gravity (9.81 m/s²)
h is the height of the ramp (1.60 m)
At the bottom of the ramp, all of the block's potential energy is converted into kinetic energy:
KE = 1/2mv^2
Setting the initial potential energy equal to the final kinetic energy:
mgh = 1/2mv^2
3. Solve for the acceleration:
From the conservation of energy equation:
a = gh = (9.81 m/s²) (1.60 m) = 15.696 m/s²
4. Apply the kinematic equation:
2.96^2 = 0 + 2 (15.696 m/s²) (1.60 m)
v^2 = 49.152 m²/s²
5. Calculate the angle of the ramp:
The angle of the ramp (θ) can be found using the following relation:
a = gsin(θ)
15.696 m/s² = 9.81 m/s² * sin(θ)
sin(θ) = 1.599
θ = sin^(-1) (1.599) = 93.3°
Therefore, the angle between the ramp and the horizontal is approximately 93.3°.
Complete question:
A 8.50-kg block of ice, released from rest at the top of a 1.60-m-long frictionless ramp, slides downhill, reaching a speed of 2.96 m/s at the bottom. What Is The Angle Between The Ramp And The Horizontal?