Acceleration Due to Applied Force
Once the tennis ball has left the racquet, it is actually a projectile, which means that it is not accelerating forward. The only time the ball is accelerating forward is when it is in contact with the racquet, because during this time there is an unbalanced force acting on it (see Newton’s Second Law). When a force is applied to the back of the ball, it is considered an unbalanced force because there is no force pushing back at it the other way and causing it to remain at a constant velocity, so it accelerates. After that, there is nothing pushing it forwards or backwards, since air resistance was minimal, so we know that it is no longer being accelerated. Normally, if the ball were starting at rest, we could use just one calculation to figure out its acceleration, but in this case, the ball was already moving towards the ground, so we will need to find the horizontal and vertical components of the acceleration first, in order to find the actual acceleration.
First, we can find the horizontal acceleration by using the horizontal velocity before contact and the horizontal velocity after contact, as well as the amount of time the ball was on the racquet for:
First, we can find the horizontal acceleration by using the horizontal velocity before contact and the horizontal velocity after contact, as well as the amount of time the ball was on the racquet for:
Therefore ball accelerates horizontally at a rate of 4.1x10^3m/s^2 while it is in contact with the racquet.
Next, we need to find the vertical acceleration of the ball. This will be a little bit more tricky since we will need to find the speed the ball was traveling at when it came into contact with the racquet We know the ball started at a height of 1.21m, and we know the acceleration due to gravity. We also know the initial velocity was zero. So to find the final velocity before the ball was hit, we must use the equation involving initial velocity, final velocity, acceleration, and displacement (we cannot use one involving time since we do not know the amount of time the ball was falling for):
Next, we need to find the vertical acceleration of the ball. This will be a little bit more tricky since we will need to find the speed the ball was traveling at when it came into contact with the racquet We know the ball started at a height of 1.21m, and we know the acceleration due to gravity. We also know the initial velocity was zero. So to find the final velocity before the ball was hit, we must use the equation involving initial velocity, final velocity, acceleration, and displacement (we cannot use one involving time since we do not know the amount of time the ball was falling for):
Therefore the vertical velocity before the ball is hit it 2.1m/s [DWN]
Now that we have this information, we can use the same steps as we did to find the horizontal acceleration:
Now that we have this information, we can use the same steps as we did to find the horizontal acceleration:
Therefore the vertical acceleration is 1.9x10^3 m/s [UP]
Finally, we can use Pythagorean theorem to solve for the actual acceleration caused by the racquet:
Finally, we can use Pythagorean theorem to solve for the actual acceleration caused by the racquet:
Therefore the racquet causes the ball to accelerate at a rate of 4.5x10^3 m/s^2 [18º above the horizontal]
It is easy for us to see that this acceleration occurs because the ball starts to move. If the ball were to not accelerated, it would have simply continued to fall straight to the floor, but since it started to move forward, we know acceleration occurred.
It is easy for us to see that this acceleration occurs because the ball starts to move. If the ball were to not accelerated, it would have simply continued to fall straight to the floor, but since it started to move forward, we know acceleration occurred.
Acceleration Due to Gravity
Once the ball has left the racquet, the only force acting on it is gravity, a force excerpted by the earth on anything on and around it. The force of gravity accelerates the ball vertically at a rate of approximately 9.8m/s^2, and is what causes the ball to fall back towards the ground. This is why we know it is occurring. If it were not, the ball would continue upwards and never come back down, but since it did fall back to earth, we know gravity was acting on it and accelerating it.