Particle motion on a vertical surface

When a particle is dropped vertically on a surface, it bounces back 25% of its original height. Let's assume the initial height of the particle is h.

The particle will bounce back to a height of 0.25h after the first bounce. On the second bounce, it will reach a height of 0.25 * 0.25h = 0.0625h. This pattern continues, and the height after each bounce can be represented as a geometric series:

h + 0.25h + (0.25)^2 * h + (0.25)^3 * h + ...

This is an infinite geometric series with a common ratio of 0.25. The sum of an infinite geometric series can be calculated using the formula:

Sum = a / (1 - r)

where a is the first term and r is the common ratio. In this case, the first term is h and the common ratio is 0.25. Plugging in these values, we can find the sum of the series:

Sum = h / (1 - 0.25) = h / 0.75 = 4h/3

Therefore, the total distance traveled by the particle before it starts sliding on the surface is 4h/3.

Particle motion when thrown at an angle

When the particle is thrown at an angle, it falls 10m away and keeps on bouncing. We can assume that the particle follows a parabolic trajectory due to the combination of its initial horizontal velocity and the downward force of gravity.

The horizontal distance traveled by the particle is independent of its vertical motion. Let's assume the initial horizontal velocity of the particle is v.

Using the equation of motion for horizontal motion:

Distance = Velocity * Time

The time taken by the particle to travel 10m horizontally can be calculated as:

10m = v * Time

Time = 10m / v

The vertical motion of the particle can be analyzed separately. The particle will follow a projectile motion, and its vertical displacement can be calculated using the equation:

Displacement = Initial Vertical Velocity * Time + (1/2) * Acceleration * Time^2

Since the particle is thrown vertically, the initial vertical velocity is 0. The only vertical force acting on the particle is gravity, which causes an acceleration of -9.8 m/s^2.

Substituting the value of time from the horizontal motion into the equation for vertical displacement:

Displacement = (1/2) * (-9.8 m/s^2) * (10m / v)^2

Simplifying the equation:

Displacement = -4.9 * (100m^2 / v^2)

The particle will keep bouncing until its vertical displacement becomes zero. In this case, the total displacement of the particle can be calculated as the sum of the horizontal and vertical displacements:

Total Displacement = 10m + (-4.9 * (100m^2 / v^2))

Total distance traveled by the particle before it starts sliding

The total distance traveled by the particle before it starts sliding can be calculated by finding the sum of the distances traveled during each bounce.

Since the particle keeps bouncing indefinitely, the total distance can be represented as an infinite series:

Total Distance = 10m + 2 * (10m * 0