In another interaction, a plasticine sphere of mass 0.30 kg is dropped 171 (c) from a height of 5.00 m and sticks to the ground. Deduce the momentum of the sphere just before impact. (i) (ii) State and explain the transfer of energy and momentum after impact.
(i) To deduce the momentum of the sphere just before impact, we need to use the formula:
Momentum (p) = mass (m) x velocity (v)
Given that the mass of the sphere is 0.30 kg, we need to find its velocity just before impact. We can use the equation for the velocity of an object in free fall:
v = √(2gh)
where g is the acceleration due to gravity (9.8 m/s^2) and h is the height (5.00 m). Plugging in the values:
v = √(2 * 9.8 * 5.00) = √(98) = 9.90 m/s
Now we can calculate the momentum:
Momentum (p) = 0.30 kg * 9.90 m/s = 2.97 kg*m/s
Therefore, the momentum of the sphere just before impact is 2.97 kg*m/s.
(ii) After the impact, the energy and momentum transfer can be explained as follows:
- Energy transfer: The potential energy of the sphere before it is dropped is converted into kinetic energy as it falls. This kinetic energy is then transferred to the ground upon impact. Some of the energy is dissipated in the form of sound and heat due to the deformation of the sphere and the ground.
- Momentum transfer: Before impact, the sphere has momentum due to its velocity. Upon impact, the sphere sticks to the ground, resulting in a sudden change in momentum. The momentum of the sphere is transferred to the ground, causing the ground to experience an equal and opposite momentum change. This transfer of momentum is a result of the conservation of momentum principle, which states that the total momentum of a system remains constant unless acted upon by external forces.
Overall, the energy is transferred from potential energy to kinetic energy and then partly dissipated, while the momentum of the sphere is transferred to the ground.
To deduce the momentum of the plasticine sphere just before impact, we can use the equation:
Momentum (p) = mass (m) x velocity (v)
Given:
Mass of the sphere (m) = 0.30 kg
Height of the sphere (h) = 5.00 m
Angle of the drop (θ) = 171°
Step 1: Determine the initial velocity of the sphere just before impact.
To find the velocity of the sphere, we need to use the concept of conservation of mechanical energy. The initial potential energy of the sphere at a height of 5.00 m is equal to the final kinetic energy just before impact.
Potential Energy (PE) = mgh
where g is the acceleration due to gravity (approximately 9.8 m/s²)
PE = 0.30 kg x 9.8 m/s² x 5.00 m
PE = 14.7 J
Kinetic Energy (KE) = (1/2) mv²
Using the principle of conservation of mechanical energy:
PE = KE
14.7 J = (1/2) x 0.30 kg x v²
v² = (2 x 14.7 J) / 0.30 kg
Step 2: Calculate the velocity of the sphere just before impact.
v² = (2 x 14.7 J) / 0.30 kg
v² = 98 J / 0.30 kg
v² = 326.67 m²/s
Taking the square root of both sides:
v = √326.67 m²/s
v ≈ 18.07 m/s
So, the velocity of the sphere just before impact is approximately 18.07 m/s.
Step 3: Calculate the momentum of the sphere just before impact.
Momentum (p) = mass (m) x velocity (v)
p = 0.30 kg x 18.07 m/s
p ≈ 5.42 kg·m/s
Thus, the momentum of the plasticine sphere just before impact is approximately 5.42 kg·m/s.
After the impact, there is a transfer of energy and momentum.
(i) Transfer of Energy:
Some of the kinetic energy of the plasticine sphere is converted into other forms of energy, such as heat and sound, upon impact with the ground. This is known as the principle of conservation of energy. The total energy of the system remains constant, but its form changes.
(ii) Transfer of Momentum:
After impact, the momentum of the plasticine sphere is transferred to the ground. Momentum is conserved in a collision, according to the principle of conservation of momentum. Therefore, the momentum of the sphere before impact will equal the momentum of the ground (and any other objects involved) after impact.
In this case, since the plasticine sphere sticks to the ground after impact, the momentum of the sphere is transferred completely to the ground, resulting in both the sphere and the ground having the same final momentum.