### Wimbledon

Rachel Plawecki
Caiti Wunderlich
Melanie Demmer
1st Hour

We originally wanted to find the relationship between the angle of a tennis racquet on a serve and the distance the ball went. However, upon shooting the video, we realized the video would allow us to find other data. The close-up video was shot at 500 frames/ second, and the full body shot was 250 frames/second. Using Logger Pro 3 and Tracker, we calculated the velocity, acceleration, momentum, and kinetic energy of the racquet at the height of the serve, the speed of the ball immediately after it was hit, the acceleration of the ball, the momentum and kinetic energy of the ball, the rotation of the ball, the force on the ball at the moment of impact, and the velocity of the server's arm during the upward motion of the serve.

Materials:
Tennis racquet, tennis ball, high speed camera, scale pole, Tracker Program

Mass of objects: Tennis ball 57g and racquet 300g

Velocity, Acceleration, Momentum, and Kinetic Energy of the Racquet
• The table above is the velocity, acceleration, momentum, and kinetic energy of the racquet from 0.5s before the point of contact to 1.0s after. The velocity of the racquet remains about the same until 3.2s when it steadily decreases. This means the motion of serving is very fluid with very few stopping points. The acceleration is less steady. The momentum and kinetic energy of the racquet remain about the same.

Speed of the Ball

As shown in the table above, the distance the ball is traveling in the x direction keeps increaing, while the distance the ball travels in the y direction decreases with each interval. The velocity of the ball in the x direction is around 1.2 m/s, and the velocity of the ball in the y direction is about -0.1 m/s. The sixth column represents the magnitude of the velocity, which is fluctuating around the 1.2 m/s mark. The third row (2.5s) is clearly when the server strikes the ball since the velocity spikes from 0.5 m/s to 1.168 m/s. The angle from the horizantal at which the ball is traveling is less than 1 degree and is negative, as the ball is traveling downward from gravity.

Acceleration of the Ball

• As shown in the table above, the maximum acceleration in the x direction occurs at 2.5s, when the server strikes the ball. The acceleration in the x direction increases from 0 to 5.022 m/s/s. After .2 seconds, however, the ball's acceleration dramatically decreases, and now the only force acting on the ball is gravity. The acceleration magnitude follows the same trend, as most of the ball's action is in the x direction.

Rotation of the Ball

• This table indicates that as time increases, the angle of rotation of the ball (theta) decreases from 1.34 degrees to 0.643 degrees. This is probably due to the decreasing impact the server's hand has over time. w represents the velocity of the ball's rotation. This column indicates that as the ball strikes the racquet at 2.5 seconds, the velocity of rotation increases (from -0.4 to -.09) then decreases after the 2.7s mark. Again, a combination of air resistance and decreasing force of the racquet are probably accountable for this.
Momentum and Kinetic Energy of the Ball

• The main point we wanted to make with this table is that the momentum and kinetic energy of the ball obviously increase after the server hits the ball. At 2.5s, the momentum jumps from 0.31kgm/s to 0.666kgm/s. The kinetic energy jumps from 0.084kJ to 0.389kJ.

Force on the Ball

• Using Tracker, we added force vectors. The force from the racquet/ person was 2.57 N (vector B), and the force from gravity was .586 N (vector A). Using the vector sum tool, we calculated that the Net Force was 2.56 N at -12.4 degrees from the horizantal.
• Vector A: 9.8 m/s/s x .057kg
• Vector B: 3kg x .858 m/s/s (acceleration of the racquet at frame 25)

Arm Speed

• Using the Analyze tool, we found the average slope of the graph to be around .6m/s. This equals the arm speed of the server on the way up to the ball.
There is a precise way the ball, racquet, and arm velocities work during a serve. The racquet speed is around .8m/s and the arm speed is around .6m/s. This velocity allows the ball to hit off the racquet at a speed of 1.2m/s. The racquet is accelerating at about 1m/s/s at the point of contact with the ball, allowing the ball to initially accelerate at 5.022m/s/s. Since the racquet weighs so much more than the ball, the force on the ball is 2.56N mostly in the horizontal direction. The faster the server's arm travels, the faster the ball will go and the higher the ball's initial acceleration will be. The more the racquet weighs, the faster the ball will go and the higher the ball's initial acceleration will be. The more force the ball is struck with, the more momentum and kinetic energy the ball will have. The server who does this with the most accuracy will have the most powerful serve.

THE WAY IT'S SUPPOSED TO BE DONE

#### Roger Federer Serve

č
TennisClose.mov
(1673k)
Steve Dickie,
May 16, 2009, 5:51 AM