A ball with mass m = 0.210 kg and kinetic energy K1 = 2.97 J collides elastically with a second ball of the same mass that is initially at rest. After the collision, the first ball moves away at an angle of = 30.6° with respect to the hori- zontal, as shown in the figure. What is the kinetic energy of the first ball after the collision?
HELP!
Please step by step
I only got the the first V of M1 that it is 5.31 m/s :/
To solve this problem and find the kinetic energy of the first ball after the collision, we need to use the principles of conservation of momentum and conservation of kinetic energy in an elastic collision. Here's a step-by-step guide:
Step 1: Determine the initial velocity of the first ball (m1) using the given kinetic energy (K1). The formula for kinetic energy is K = (1/2)mv^2, where K is the kinetic energy, m is the mass, and v is the velocity. Rearranging the formula, we get v = sqrt((2K)/m).
Given: m1 = 0.210 kg, K1 = 2.97 J.
v1 = sqrt((2 * 2.97) / 0.210) = 5.31 m/s (as you correctly calculated)
Step 2: The second ball (m2) is initially at rest, so its initial velocity (v2) is 0 m/s.
Step 3: Calculate the total initial momentum, which is the sum of the individual momenta of the two balls.
Initial momentum (p_initial) = m1 * v1 + m2 * v2
Since m2 * v2 is 0, the equation becomes:
p_initial = m1 * v1
p_initial = (0.210 kg) * (5.31 m/s)
p_initial = 1.1151 kg·m/s
Step 4: After the collision, both balls move away from each other. The first ball moves away at an angle of 30.6° with respect to the horizontal. Let's denote its final velocity as (v1f) and the angle as (θ).
Step 5: We can resolve the final velocity into horizontal and vertical components. The horizontal component can be found by multiplying the magnitude of the velocity by the cosine of the angle, and the vertical component can be found by multiplying the magnitude of the velocity by the sine of the angle.
v1f_horiz = v1f * cos(θ)
v1f_vert = v1f * sin(θ)
Step 6: Since this is an elastic collision, the total momentum before the collision should be equal to the total momentum after the collision.
p_initial = p_final
m1 * v1 + m2 * v2 = m1 * v1f_horiz + m2 * v2f
Since m2 was initially at rest, the equation becomes:
m1 * v1 = m1 * v1f_horiz
v1 = v1f_horiz
(1.1151 kg·m/s) = (0.210 kg) * v1f_horiz
Step 7: Solve for v1f_horiz.
v1f_horiz = (1.1151 kg·m/s) / (0.210 kg)
v1f_horiz = 5.31 m/s
Step 8: Now, we have the horizontal component of the final velocity. To find the kinetic energy of the first ball after the collision, we can square this value and multiply it by half of the mass.
K1f = (1/2) * m1 * (v1f_horiz)^2
K1f = (1/2) * (0.210 kg) * (5.31 m/s)^2
K1f = 3.2869 J
Therefore, the kinetic energy of the first ball after the collision is approximately 3.287 J.