A 751 kg block is pushed on the slope of a 30◦
frictionless inclined plane to give it an initial
speed of 50 cm/s along the slope when the
block is 1.3 m from the bottom of the incline.
The acceleration of gravity is 9.8 m/s^2. What is the speed of the block at the bottom for the plane?
h = 1.3*sin30 = 0.65 m.
Vo = 50cm = 0.50 m.
V = Speed at bottom of the ramp.
V^2 = Vo^2 + 2g*h.
V^2 = 0.50^2 + 19.8*0.65 =
V = 3.60 m/s.
To find the speed of the block at the bottom of the inclined plane, we can use the principles of conservation of energy.
Let's find the potential energy (PE) and kinetic energy (KE) of the block at the given point.
Given:
Mass of the block (m) = 751 kg
Angle of the inclined plane (θ) = 30 degrees
Initial speed of the block (v) = 50 cm/s = 0.5 m/s
Distance from the bottom of the incline (d) = 1.3 m
Acceleration due to gravity (g) = 9.8 m/s^2
First, let's calculate the height of the block from the bottom of the incline to its current position.
Using trigonometry, we can determine the height (h) as follows:
h = d * sin(θ)
h = 1.3 m * sin(30 degrees)
h = 1.3 m * 0.5
h = 0.65 m
Next, let's find the initial potential energy of the block.
PE_initial = m * g * h
PE_initial = 751 kg * 9.8 m/s^2 * 0.65 m
PE_initial = 4711.7 J
Now, let's find the initial kinetic energy of the block.
KE_initial = 0.5 * m * v^2
KE_initial = 0.5 * 751 kg * (0.5 m/s)^2
KE_initial = 0.5 * 751 kg * 0.25 m^2/s^2
KE_initial = 94.125 J
Considering that there is no friction, the total mechanical energy (E) at any point along the inclined plane remains constant.
E_initial = PE_initial + KE_initial
E_initial = 4711.7 J + 94.125 J
E_initial = 4805.825 J
Now, let's find the final kinetic energy of the block at the bottom of the inclined plane.
KE_final = E_initial
Since there is no potential energy at the bottom (PE_final = 0), the total energy is entirely kinetic.
KE_final = 4805.825 J
Finally, let's calculate the final speed of the block at the bottom using the final kinetic energy.
KE_final = 0.5 * m * v_final^2
v_final^2 = (2 * KE_final) / m
v_final^2 = (2 * 4805.825 J) / 751 kg
v_final^2 = 16.1072 m^2/s^2
v_final = √(16.1072 m^2/s^2)
v_final ≈ 4.01 m/s
Therefore, the speed of the block at the bottom of the inclined plane is approximately 4.01 m/s.
To find the speed of the block at the bottom of the inclined plane, we can use the principles of conservation of energy and kinematics. Here's how we can solve for it:
1. First, let's understand the initial and final positions of the block. The block starts at a height of 1.3 m from the bottom of the incline and moves down the slope to the bottom of the incline.
2. The gravitational potential energy of the block at the initial position is given by the formula: PE = mgh, where m is the mass of the block (751 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height of the block above the bottom of the incline (1.3 m).
So, the initial potential energy (PE_initial) of the block is: PE_initial = 751 kg * 9.8 m/s² * 1.3 m.
3. The initial kinetic energy of the block is given by: KE = 1/2 * m * v_initial^2, where m is the mass of the block (751 kg) and v_initial is the initial speed of the block along the slope (50 cm/s). We need to convert the speed to meters per second (m/s).
So, the initial kinetic energy (KE_initial) of the block is: KE_initial = 1/2 * 751 kg * (50 cm/s)^2.
4. Since the inclined plane is frictionless, there is no work done against friction. Therefore, the total mechanical energy of the block is conserved during the motion. Hence, the sum of initial potential energy and initial kinetic energy is equal to the sum of final potential energy (PE_final) and final kinetic energy (KE_final).
So, PE_initial + KE_initial = PE_final + KE_final.
5. At the bottom of the incline, the block will have no height above the bottom of the incline. Therefore, the final potential energy (PE_final) of the block is zero.
So, PE_final = 0.
6. We want to find the final kinetic energy (KE_final), which will help us determine the speed of the block at the bottom of the incline.
Rearranging the conservation of energy equation, we get: KE_final = PE_initial + KE_initial - PE_final.
Substituting the values, we get: KE_final = (1/2 * 751 kg * (50 cm/s)^2) + (751 kg * 9.8 m/s² * 1.3 m).
7. Simplifying the equation and calculating, we can find the final kinetic energy (KE_final). Then, we can calculate the speed of the block at the bottom by using the formula: KE_final = 1/2 * m * v_final^2, where m is the mass of the block (751 kg) and v_final is the speed of the block at the bottom.
Solving for v_final, we can rearrange the equation to: v_final = √(2 * KE_final / m).
Substituting the values of KE_final and m, we can calculate the speed of the block at the bottom of the incline.