A ball rolls without slipping at 6 m/s off an incline of 25 degrees. The ball falls off the incline and enters a hole 3 meters below the incline. What is the horizontal distance between the bottom of the incline and the hole?

the horizontal speed is 6cos25° = 5.44 m/s

How long does it take to fall 3 meters?

4.9t^2 = 3
t = 0.78

Now you know the horizontal distance it went as it fell:

0.78 * 5.44 = 4.26 meters

To find the horizontal distance between the bottom of the incline and the hole, we can use the equations of motion and the principle of conservation of energy.

Let's break down the problem into two parts: the motion on the incline and the free fall.

1. Motion on the Incline:
The ball is rolling without slipping, which means that the velocity of the ball can be related to its angular velocity and radius of the ball. The equation for the velocity of a rolling ball is given by:
v = ωr
where v is the linear velocity, ω is the angular velocity, and r is the radius of the ball.

Given that the ball is rolling at 6 m/s, we need to find the angular velocity. We can use the fact that the linear velocity is related to the angular velocity as follows:
v = ωr
6 m/s = ωr
Since the ball is rolling without slipping, the radius of the ball does not change. Therefore, we can further simplify the equation as:
ω = 6 m/s / r

Next, let's determine the acceleration of the ball on the incline. The acceleration can be calculated using Newton's second law:
F = ma
where F is the net force acting on the ball, m is the mass of the ball, and a is the acceleration.

The net force can be resolved into two components: the component due to gravity parallel to the incline, and the component due to the normal force perpendicular to the incline. Since the ball is rolling without slipping, the frictional force is zero.

The component of gravity parallel to the incline is given by:
F_parallel = mg sin(θ)
where m is the mass of the ball, g is the acceleration due to gravity, and θ is the angle of the incline.

Substituting the value of F_parallel in Newton's second law equation, we have:
mg sin(θ) = ma
canceling out mass "m" on both sides,
g sin(θ) = a

The ball rolls down an incline at an angle of 25 degrees, so we can calculate the acceleration:
a = g sin(25°)

2. Free Fall:
After leaving the incline, the ball falls freely under the influence of gravity. The vertical distance the ball falls can be calculated using kinematic equations:
h = (1/2)gt^2
where h is the vertical distance, g is the acceleration due to gravity, and t is the time taken.

The vertical distance h is given as 3 meters. We need to find the time taken for the ball to fall from the incline to the hole.

We can determine the time by using the equation:
h = (1/2)gt^2
3 m = (1/2)(9.8 m/s^2)t^2
Solving for t, we get:
t = √(2 * 3 m / 9.8 m/s^2)

Now, we know the time it takes for the ball to fall. During this time, the horizontal distance covered by the ball can be found using the equation:
horizontal distance = velocity x time

Substituting the values:
horizontal distance = 6 m/s * √(2 * 3 m / 9.8 m/s^2)

By calculating this expression, we can find the horizontal distance between the bottom of the incline and the hole.