A billiard ball that is initially at rest is given a sharp blow by a cue stick. The force is horizontal and is applied at a distance h = 2R/3 below the centerline, as shown in the figure below. The speed of the ball just after the blow is v0 and the coefficient of kinetic friction between the ball and the billiard table is muk
There is no "figure below" and there is also no question.
Well, looks like this billiard ball is about to have a wild ride! Let's break it down.
First off, the force being applied horizontally by the cue stick is like a firm handshake saying, "Let's get this ball rolling!" But, instead of applying it at the centerline, they decided to go for a more eccentric location, 2R/3 below the centerline. Maybe they wanted to show off their fancy shot.
Now, let's talk speed. After the blow, this ball will be going at a speed of v0. It's like saying, "Hey, gravity, I'm about to defy you!" It's the ball's way of saying, "Catch me if you can!"
But alas, there's the coefficient of kinetic friction (muk), the party pooper of physics. It's like the ball's old nemesis, trying to slow it down for no good reason. It's the "stop right there, ball" of the billiard world.
So, in conclusion, this ball experiences a fancy, off-center hit, shoots off with a specific speed, and then faces the force of friction like a speed bump on the road of life. Just remember, even billiard balls need a little drama in their lives.
To find the final speed of the billiard ball after the blow, we need to consider the forces acting on the ball and apply Newton's laws of motion.
1. Identify the forces acting on the ball:
- The force applied by the cue stick, which is horizontal.
- The normal force exerted by the table, which is perpendicular to the table surface.
- The force of kinetic friction between the ball and the table.
2. Determine the net force acting on the ball:
The net force is the vector sum of all the forces acting on the ball.
- The horizontal component of the force applied by the cue stick will cause the ball to accelerate.
- The vertical component of the force applied by the cue stick will contribute to the normal force, but will not affect the horizontal motion of the ball.
- The normal force will balance the vertical component of the force applied by the cue stick.
- The force of kinetic friction will oppose the motion of the ball.
Since the ball is initially at rest, the net force is given by
net force = force applied by cue stick - force of kinetic friction
3. Calculate the acceleration of the ball:
From Newton's second law, F = ma, where F is the net force and a is the acceleration.
Rearranging the equation and substituting the known values, we have
a = (net force) / m,
where m is the mass of the ball.
4. Calculate the distance traveled by the ball:
Using the formula s = ut + (1/2)at^2, where s is the distance, u is the initial velocity (0 m/s), t is the time, and a is the acceleration.
Since the ball starts from rest, the equation simplifies to s = (1/2)at^2.
5. Determine the time it takes for the ball to reach the final speed:
Use the formula v = u + at, where v is the final velocity, u is the initial velocity (0 m/s), a is the acceleration, and t is the time.
6. Calculate the final speed of the ball:
The final speed is given by v = u + at, where u = 0 m/s and a is the acceleration.
That's the step-by-step process to find the final speed of the billiard ball after the blow.
To find the subsequent motion of the billiard ball after the blow, we need to consider the forces acting on the ball and the resulting acceleration.
1. Let's start by identifying the forces acting on the ball. The force exerted by the cue stick is the initial driving force, which is horizontal. The weight of the ball acts vertically downward. Additionally, there is a frictional force opposing the ball's motion due to the coefficient of kinetic friction.
2. The key concept to understand here is that the horizontal force provided by the cue stick induces both linear and rotational motion in the ball. The linear motion occurs along the horizontal direction, while the rotational motion causes the ball to spin.
3. Now, let's analyze the forces in more detail. The driving force and the frictional force act horizontally, opposing each other. The weight of the ball acts vertically downward. The vertical component of the weight does not affect the horizontal motion of the ball.
4. To determine the net force acting horizontally, we need to subtract the frictional force from the driving force. The frictional force can be calculated by multiplying the coefficient of kinetic friction (muk) with the normal force, which is equal to the weight of the ball.
5. The net force acts as the driving force for the horizontal motion of the ball. Using Newton's second law (F = ma), where F is the net force and a is acceleration, we can calculate the resulting acceleration.
6. With the acceleration value, you can find the subsequent velocity (speed) of the ball using the equation v = u + at, where v is the final velocity after the blow, u is the initial velocity (which is zero since the ball is initially at rest), a is the acceleration, and t is the time.
It's important to note that the rotational motion induced by the blow may also affect the subsequent motion of the ball. To fully analyze the ball's trajectory, you would need to consider the angular momentum and torque acting on the ball due to the applied force.