A 80 kg object slides at a 20 degree plane downward at 5m/s. An object 2 collides with object 1 by falling from 7 meter height and they stick together. Final speed of objects?
I know the speeds would be the same. Would momentum equation work to solve this? The perfect inelastic equation does not have height. How do I incorporate height to get final speed of objects.
To solve this problem, you can use the law of conservation of momentum and the principle of conservation of mechanical energy.
First, let's find the initial speed of object 2 before the collision. Since it falls from a height of 7 meters, you can use the formula for gravitational potential energy to determine its initial speed:
Potential Energy (PE) = mass (m) * gravity (g) * height (h)
The potential energy is converted to kinetic energy as the object falls, so we can say:
Potential Energy (PE) = Kinetic Energy (KE)
So,
mgh = (1/2) * m * v2^2
Where m is the mass of object 2, g is the acceleration due to gravity (approximately 9.8 m/s^2), h is the height (7 meters), and v2 is the initial velocity of object 2.
Simplifying the equation, we get:
gh = (1/2) * v2^2
Substituting the known values:
(9.8 m/s^2) * (7 m) = (1/2) * v2^2
Solving for v2, we find:
v2 = sqrt((9.8 m/s^2) * (7 m) * 2) = sqrt(137.2) = 11.72 m/s
Now, let's use the law of conservation of momentum to find the final speed of the combined objects after the collision. The law states that the total momentum before the collision is equal to the total momentum after the collision.
Momentum (p) = mass (m) * velocity (v)
Object 1:
Initial momentum (p1_initial) = mass (80 kg) * velocity (5 m/s)
Object 2:
Initial momentum (p2_initial) = mass (unknown) * velocity (11.72 m/s)
Since the objects stick together after the collision, the combined mass (m1 + m2) will be 80 kg + mass of object 2. Let's represent the combined mass as M.
Final momentum (p_final) = combined mass (M) * final velocity (v_final)
Applying the law of conservation of momentum:
p1_initial + p2_initial = p_final
(80 kg * 5 m/s) + (mass of object 2 * 11.72 m/s) = (M) * (v_final)
Now, let's use the principle of conservation of mechanical energy to relate the height to the final speed.
Initial potential energy = Kinetic energy after the collision
mgh = (1/2) * (M) * v_final^2
Substituting the known values:
(80 kg * 9.8 m/s^2 * 7 m) = (1/2) * (M) * v_final^2
Solving for v_final, we get:
v_final = sqrt((80 kg * 9.8 m/s^2 * 7 m) / ((1/2) * (M)))
Now, substitute the expression for M from the momentum equation:
v_final = sqrt((80 kg * 9.8 m/s^2 * 7 m) / ((1/2) * (80 kg + mass of object 2)))
Finally, you can solve for the final speed of the objects by substituting the mass of object 2 into the equation and calculating the square root of the expression.