In an ice show a 40.0 kg skater leaps into the air and is caught by an initially stationary 65.0 kg skater.
(a) What is their final velocity assuming negligible friction and that the leaper's original horizontal velocity was 4.00 m/s?
(b) How much kinetic energy is lost?
a) The final velocity of the two skaters is 4.00 m/s.
b) The kinetic energy lost is equal to the difference in the kinetic energies of the two skaters before and after the leap. The initial kinetic energy of the 65.0 kg skater is 0, and the initial kinetic energy of the 40.0 kg skater is 160 J. The final kinetic energy of the two skaters is 80 J. Therefore, the kinetic energy lost is 80 J.
To find the final velocity of the skaters after the leaper is caught and the amount of kinetic energy lost, we can apply the principles of conservation of momentum and conservation of kinetic energy.
(a) To find the final velocity of the skaters, we can use the conservation of momentum:
The initial momentum of the system is equal to the final momentum of the system, assuming no external forces are acting on it.
Initial momentum = Final momentum
The initial momentum can be calculated by considering the momentum of each skater separately:
Momentum of the leaper (before catch) = mass of leaper × initial velocity of leaper
= 40.0 kg × 4.00 m/s
Momentum of the stationary skater (before catch) = mass of stationary skater × 0 (as initially stationary)
= 65.0 kg × 0
The final momentum is the sum of the momenta of both skaters after the catch. Let's denote the final velocity of the skaters as v_f.
Final momentum = (mass of leaper + mass of stationary skater) × final velocity of skaters
= (40.0 kg + 65.0 kg) × v_f
Setting the initial momentum equal to the final momentum, we have:
40.0 kg × 4.00 m/s + 65.0 kg × 0 = (40.0 kg + 65.0 kg) × v_f
Simplifying the equation:
160 kg·m/s = 105 kg × v_f
Dividing both sides of the equation by 105 kg:
v_f = 160 kg·m/s / 105 kg
Finally, calculating the final velocity:
v_f = 1.52 m/s
Therefore, the final velocity of the skaters after the catch is approximately 1.52 m/s.
(b) To find the amount of kinetic energy lost, we can subtract the final kinetic energy from the initial kinetic energy.
The initial kinetic energy of the leaper can be calculated using the formula:
Initial kinetic energy of leaper = 1/2 × mass of leaper × (initial velocity of leaper)^2
Initial kinetic energy of leaper = 1/2 × 40.0 kg × (4.00 m/s)^2
The final kinetic energy of the skaters can be calculated using the formula:
Final kinetic energy = 1/2 × (mass of leaper + mass of stationary skater) × (final velocity)^2
Final kinetic energy = 1/2 × (40.0 kg + 65.0 kg) × (1.52 m/s)^2
The amount of kinetic energy lost is given by the difference between the initial and final kinetic energy:
Kinetic energy lost = Initial kinetic energy of leaper - Final kinetic energy
Substituting the values, we can calculate the kinetic energy lost.
To find the final velocity of the skaters, we can use the principle of conservation of momentum. According to this principle, the total momentum before the catch should be equal to the total momentum after the catch.
Let's break down the problem step-by-step:
Step 1: Find the initial momentum of the leaper and the stationary skater.
The momentum (p) of an object is given by the equation: p = mass (m) × velocity (v).
For the leaper:
Mass (m1) = 40.0 kg
Velocity (v1) = 4.00 m/s
The initial momentum of the leaper (p1) is: p1 = m1 × v1
For the stationary skater:
Mass (m2) = 65.0 kg
Velocity (v2) = 0 m/s (as the skater is initially stationary)
The initial momentum of the stationary skater (p2) is: p2 = m2 × v2
Step 2: Find the total initial momentum.
The total initial momentum (p_initial) is the sum of the initial momentum of the leaper and the stationary skater: p_initial = p1 + p2
Step 3: Find the total mass of the skaters.
The total mass (m_total) is the sum of the mass of the leaper and the mass of the stationary skater: m_total = m1 + m2
Step 4: Use the conservation of momentum to find the final velocity.
According to the conservation of momentum, the total momentum after the catch (p_final) should be equal to the total initial momentum (p_initial):
p_final = p_initial
Using the equation p = m × v, we can rewrite this as:
(m1 + m2) × v_final = m1 × v1 + m2 × v2
Rearranging the equation to solve for the final velocity (v_final):
v_final = (m1 × v1 + m2 × v2) / (m1 + m2)
Now, let's substitute the values into the equation to find the final velocity:
v_final = (40.0 kg × 4.00 m/s + 65.0 kg × 0 m/s) / (40.0 kg + 65.0 kg)
Therefore, the final velocity of the skaters when the leaper is caught is:
v_final = 1.85 m/s (rounded to two decimal places)
Now, let's move on to part (b) to find the amount of kinetic energy lost.
Step 5: Find the initial kinetic energy of the leaper.
The kinetic energy (KE) of an object is given by the equation: KE = (1/2) × mass (m) × velocity^2 (v^2).
For the leaper:
Mass (m1) = 40.0 kg
Velocity (v1) = 4.00 m/s
The initial kinetic energy of the leaper (KE1) is: KE1 = (1/2) × m1 × v1^2
Step 6: Find the final kinetic energy of the skaters.
The final kinetic energy (KE_final) of the skaters is given by the equation: KE_final = (1/2) × total mass (m_total) × final velocity^2 (v_final^2).
The total mass (m_total) = m1 + m2
Let's substitute the values into the equation to find the final kinetic energy:
KE_final = (1/2) × (40.0 kg + 65.0 kg) × (1.85 m/s)^2
Step 7: Find the amount of kinetic energy lost.
The amount of kinetic energy lost (ΔKE) is the difference between the initial kinetic energy of the leaper and the final kinetic energy of the skaters:
ΔKE = KE1 - KE_final
Substitute the values into the equation to find the amount of kinetic energy lost:
ΔKE = KE1 - [(1/2) × (40.0 kg + 65.0 kg) × (1.85 m/s)^2]
By evaluating the subtraction, you can find the amount of kinetic energy lost.