A 5.3 g coin sliding to the right at 26.8 cm/s makes an elastic head-on collision with a 15.9 g coin that is initially at rest. After the collision, the 5.3 g coin moves to the left at 13.4 cm/s.
(a) Find the final velocity of the other coin.
cm/s
(b) Find the amount of kinetic energy transferred to the 15.9 g coin.
J
To find the final velocity of the other coin, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.
First, let's calculate the momentum before and after the collision for each coin:
Momentum of the 5.3 g coin before the collision:
mass × velocity = (5.3 g) × (26.8 cm/s)
Note: We need to convert the mass from grams to kilograms and the velocity from cm/s to m/s to get the correct unit for momentum in kg·m/s.
The mass is 5.3 g = 0.0053 kg.
The velocity is 26.8 cm/s = 0.268 m/s.
So the momentum before the collision for the 5.3 g coin is:
0.0053 kg × 0.268 m/s = 0.0014204 kg·m/s.
Momentum of the 15.9 g coin before the collision:
Since the 15.9 g coin is initially at rest, its momentum before the collision is 0 kg·m/s.
Now, let's find the velocities of both coins after the collision:
Momentum of the 5.3 g coin after the collision:
Using the same formula as before:
mass × velocity = (5.3 g) × (-13.4 cm/s)
Note: The velocity of the 5.3 g coin after the collision is in the opposite direction, so we use a negative sign.
Converting the mass and velocity to the correct units:
Mass = 0.0053 kg
Velocity = -0.134 m/s (13.4 cm/s = 0.134 m/s)
Momentum of the 5.3 g coin after the collision:
0.0053 kg × (-0.134 m/s) = -0.0007122 kg·m/s.
Now, let's find the momentum of the 15.9 g coin after the collision, represented by P2:
P2 (momentum of the 15.9 g coin) = ??? kg·m/s.
Since momentum is conserved, we equate the total momentum before the collision to the total momentum after the collision:
Total momentum before = Total momentum after
(0.0014204 kg·m/s) + 0 kg·m/s = -0.0007122 kg·m/s + P2.
Simplifying this equation:
0.0014204 kg·m/s = P2 - 0.0007122 kg·m/s.
To find P2, we can rearrange this equation:
P2 = 0.0014204 kg·m/s + 0.0007122 kg·m/s.
Calculating the value of P2:
P2 = 0.0021326 kg·m/s.
Therefore, the final momentum of the 15.9 g coin after the collision is equal to 0.0021326 kg·m/s.
Now, let's move on to part (b) and find the amount of kinetic energy transferred to the 15.9 g coin.
To calculate the kinetic energy transferred, we need to find the difference in kinetic energy of the 15.9 g coin before and after the collision, as kinetic energy is dependent on mass and velocity.
Kinetic energy before the collision:
The 15.9 g coin is initially at rest, so its kinetic energy before the collision is 0.
Kinetic energy after the collision:
The kinetic energy after the collision can be calculated using the formula: KE = (1/2) × mass × velocity^2.
Mass = 15.9 g = 0.0159 kg.
Velocity = P2 / mass (Note: P2 is the momentum of the 15.9 g coin after the collision.)
Substituting the values into the formula:
KE = (1/2) × 0.0159 kg × (P2/mass)^2.
Calculating the value:
KE = (1/2) × 0.0159 kg × (0.0021326 kg·m/s / 0.0159 kg)^2.
Simplifying:
KE = 5.36695 × 10^(-5) J.
Therefore, the amount of kinetic energy transferred to the 15.9 g coin is approximately 5.36695 × 10^(-5) J.