Physics students are performing an experiment and slide a hockey puck off a horizontal desk that is 1.2m high. The initial speed of the puck is 1.5m/s. Determine the final velocity(including the angle) the puck has the moment it hits the ground.
To solve this problem, we need to consider the conservation of energy.
The initial kinetic energy of the puck is given by the formula:
KEi = (1/2)mv^2
where m is the mass of the puck and v is the initial velocity.
The final potential energy of the puck is given by the formula:
PEf = mgh
where m is the mass of the puck, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the desk.
Since the energy cannot be created or destroyed, the initial kinetic energy is equal to the final potential energy:
KEi = PEf
(1/2)mv^2 = mgh
Cancelling out the mass:
(1/2)v^2 = gh
Rearranging the equation:
v^2 = 2gh
Taking the square root of both sides:
v = √(2gh)
Substituting the given values:
v = √(2 * 9.8 m/s^2 * 1.2 m)
v ≈ √(23.52 m^2/s^2)
v ≈ 4.85 m/s
The final velocity of the puck just before it hits the ground is approximately 4.85 m/s.
To determine the final velocity of the puck when it hits the ground, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if there are no external forces acting on it.
Let's assume the initial velocity of the puck is v, and its final velocity is v', and the height from which it is dropped is h.
According to the principle of conservation of mechanical energy, the initial mechanical energy of the puck (at the top of the desk) is equal to its final mechanical energy (at the ground level). The mechanical energy is the sum of kinetic energy (KE) and potential energy (PE).
At the top of the desk:
Initial PE = m * g * h (where m is the mass of the puck, g is the acceleration due to gravity, and h is the height)
At the ground level:
Final KE + Final PE = 0 (since the puck comes to rest after hitting the ground)
The initial kinetic energy (KE) is given as (1/2) * m * v^2.
To find the final velocity including the angle, we need to break it down into horizontal and vertical components.
The horizontal component of the velocity (v_x') remains unchanged, and the vertical component of the velocity (v_y') is zero, as the puck comes to rest.
Let's solve step-by-step:
Step 1: Calculate the initial potential energy (PE) at the top of the desk.
PE = m * g * h
Given:
m (mass of the puck) = ? (not provided)
g (acceleration due to gravity) = 9.8 m/s^2
h (height) = 1.2 m
Step 2: Calculate the initial kinetic energy (KE) at the top of the desk.
KE = (1/2) * m * v^2
Given:
v (initial speed) = 1.5 m/s
Step 3: Set up the conservation of mechanical energy equation.
Initial PE = Final KE + Final PE
Step 4: Solve for the final potential energy (PE) at the ground level.
Final PE = 0
Step 5: Set up the equation for solving the final kinetic energy (KE) at the ground level.
Final KE + Final PE = 0
Step 6: Solve for the final velocity including the angle.
We can assume that the final horizontal component of the velocity (v_x') remains unchanged, so it will be equal to the initial velocity (v_x' = v_x = v * cosθ).
The final vertical component of the velocity (v_y') will be zero (v_y' = 0).
Given:
v_x' = v * cosθ = v_x (horizontal component of initial velocity)
v_y' = 0 (vertical component of final velocity)
Therefore, the final velocity including the angle when the puck hits the ground is (v_x', v_y') = (v_x, v_y').
Please note that the mass of the puck is not provided in the question, so we cannot calculate the exact values for kinetic energy, potential energy, and the final velocity.
To determine the final velocity of the puck when it hits the ground, we can use the principles of projectile motion.
Step 1: Break down the initial velocity into its horizontal and vertical components:
The initial velocity of the puck can be divided into two components: horizontal and vertical. The horizontal component remains constant throughout the motion, while the vertical component changes due to the influence of gravity.
Given that the initial velocity of the puck is 1.5 m/s and assuming no horizontal acceleration, the horizontal component remains 1.5 m/s throughout the motion.
To find the vertical component, we need to calculate the initial vertical velocity. We can use the formula:
v₀y = v₀ * sin(θ)
where:
v₀y = initial vertical velocity
v₀ = initial velocity of the puck
θ = angle of projection (in this case, the angle between the horizontal and the direction of the puck's velocity)
Since the puck is sliding off the desk horizontally (without any upward or downward projection), the angle of projection is 0 degrees. Therefore, sin(θ) = sin(0) = 0.
Thus, the initial vertical velocity, v₀y, is 0 m/s.
Step 2: Calculate the time taken for the puck to fall:
To determine the time taken for the puck to fall, we can use the equation:
Δy = v₀y * t + (1/2) * g * t²
where:
Δy = change in vertical position (height of the desk)
v₀y = initial vertical velocity (0 m/s)
g = acceleration due to gravity (approximately 9.8 m/s²)
t = time taken to fall
Plugging in the values, we get:
1.2 = 0 * t + (1/2) * 9.8 * t²
1.2 = 4.9t²
Solving for t², we find:
t² = (1.2 / 4.9)
t² ≈ 0.2449
Taking the square root of both sides, we get:
t ≈ √0.2449
t ≈ 0.4949 s
Step 3: Calculate the final velocity:
Finally, we can find the final velocity by using the equation:
v = v₀ + g * t
where:
v = final velocity
v₀ = initial velocity of the puck (1.5 m/s)
g = acceleration due to gravity (9.8 m/s²)
t = time taken to fall (0.4949 s)
Plugging in the values, we have:
v = 1.5 + 9.8 * 0.4949
v ≈ 1.5 + 4.854
v ≈ 6.354 m/s
Therefore, the final velocity of the puck when it hits the ground is approximately 6.354 m/s in the vertical direction. However, since the puck slides off the desk horizontally, the final velocity in the horizontal direction remains 1.5 m/s (the initial horizontal velocity).
Hence, the final velocity (including the angle) of the puck when it hits the ground is approximately 6.354 m/s at an angle of 0 degrees (horizontal).