High speed stroboscopic photographs show that the head of a 188 g golf club is traveling at 52.5 m/s just before it strikes a 45.5g golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at 44.0 m/s. Find the speed of the golf ball just after impact
To find the speed of the golf ball just after impact, we can use the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
Given:
Mass of the golf club head (m1) = 188 g = 0.188 kg
Velocity of the golf club head before the collision (v1) = 52.5 m/s
Mass of the golf ball (m2) = 45.5 g = 0.0455 kg
Velocity of the golf ball before the collision (v2) = 0 m/s
Velocity of the golf club head after the collision (v3) = 44.0 m/s
The momentum before the collision is the momentum of the club head, and after the collision, it is the sum of the momentum of the club head and the momentum of the golf ball.
Before collision:
Momentum of the club head = m1 * v1
= 0.188 kg * 52.5 m/s
After collision:
Momentum of the club head + Momentum of the golf ball = (m1 * v3) + (m2 * v4)
Since the velocity of the golf ball just after the impact is unknown, we will denote it as v4.
Applying the law of conservation of momentum:
m1 * v1 = (m1 * v3) + (m2 * v4)
Substituting the given values:
0.188 kg * 52.5 m/s = (0.188 kg * 44.0 m/s) + (0.0455 kg * v4)
Simplifying the equation:
9.87 kg·m/s = 8.272 kg·m/s + 0.0455 kg * v4
Rearranging the equation to solve for v4:
0.0455 kg * v4 = 9.87 kg·m/s - 8.272 kg·m/s
= 1.598 kg·m/s
Dividing both sides by 0.0455 kg to solve for v4:
v4 = 1.598 kg·m/s / 0.0455 kg
≈ 35.09 m/s
Therefore, the speed of the golf ball just after impact is approximately 35.09 m/s.
To find the speed of the golf ball just after impact, we can make use of the law of conservation of momentum.
The law of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, provided no external forces are acting on the system.
In this case, we have a system consisting of the golf club and the golf ball. Before the collision, only the golf club is moving, and after the collision, both the golf club and the golf ball are moving.
Let's denote the initial velocity of the golf ball as v_ball and the final velocity of the golf ball as v'_ball. Similarly, let's denote the initial velocity of the golf club as v_club and the final velocity of the golf club as v'_club.
According to the law of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:
(m_ball * v_ball) + (m_club * v_club) = (m_ball * v'_ball) + (m_club * v'_club)
Where:
m_ball = mass of the golf ball = 45.5 g = 0.0455 kg
m_club = mass of the golf club = 188 g = 0.188 kg
v_club = initial velocity of the golf club = 52.5 m/s
v'_club = final velocity of the golf club = 44.0 m/s
Since the golf ball is at rest initially, its initial velocity (v_ball) is 0 m/s.
Substituting the known values into the equation, we get:
(0.0455 kg * 0 m/s) + (0.188 kg * 52.5 m/s) = (0.0455 kg * v'_ball) + (0.188 kg * 44.0 m/s)
0 + 9.93 kg·m/s = 0.0455 kg * v'_ball + 8.31 kg·m/s
Rearranging the equation to solve for v'_ball:
0.0455 kg * v'_ball = 9.93 kg·m/s - 8.31 kg·m/s
0.0455 kg * v'_ball = 1.62 kg·m/s
Dividing both sides by 0.0455 kg:
v'_ball = 1.62 kg·m/s / 0.0455 kg
v'_ball ≈ 35.6 m/s
Therefore, the speed of the golf ball just after impact is approximately 35.6 m/s.