Jumping on a sliding board with no friction between surfaces.

[judas_priest]

Homework Statement

You stand at the end of a long board of length L. The board rests on a frictionless frozen surface of a pond. You want to jump to the opposite end of the board. What is the minimum take-off speed v measured with respect to the pond that would allow you to accomplish that? The board and you have the same mass m.

What is v?

Homework Equations

The Attempt at a Solution

Since there is no friction, on attempt of jumping, the skateboard is going to move back some distance. How do I find the distance and relate to the velocity? Clueless about this problem

 

[barryj]

Just to be clear, I assume that you jump with a velocity and an angle with respect to the board, yes?

 

[haruspex]

Suppose you take off at speed v, relative to the pond, and angle theta to the horizontal. What will be the vertical and horizontal components of your speed? How long will you be in the air? What horizontal speed did the board get? What then is your horizontal speed relative to the board?

 

[mshmsh_2100]

sqrt(g*L/2)

 

[Nickie]

.



I would approach this problem by first considering the concept of conservation of energy. Since there is no friction, the total mechanical energy of the system (skateboard and person) will remain constant. This means that the initial kinetic energy of the person before the jump will be equal to the final kinetic energy of the person after the jump, plus any potential energy gained during the jump.

Next, I would consider the forces acting on the person during the jump. Since there is no friction, the only force acting on the person is the force of gravity pulling them down towards the ground. This means that the person's motion can be described by the equations of projectile motion.

To solve for the minimum take-off speed, I would use the equation for the horizontal distance traveled by a projectile:

x = v0 * t

Where x is the distance traveled, v0 is the initial velocity, and t is the time of flight. In this case, we want to find the minimum initial velocity (v0) that will allow the person to travel a distance of L (the length of the board).

Next, I would use the equation for the vertical motion of a projectile:

y = y0 + v0y * t + 1/2 * a * t^2

Where y is the vertical position, y0 is the initial vertical position (in this case, the height of the person on the board), v0y is the initial vertical velocity, a is the acceleration due to gravity, and t is the time of flight.

Since we want the person to land at the opposite end of the board, we can set the final vertical position (y) equal to 0. We can also set the initial vertical position (y0) equal to the height of the person on the board. This will allow us to solve for the initial vertical velocity (v0y).

Once we have the initial vertical velocity, we can use the Pythagorean theorem to find the magnitude of the initial velocity (v0):

v0 = √(v0x^2 + v0y^2)

Where v0x is the horizontal component of the initial velocity, which we found earlier using the equation for horizontal distance traveled.

In summary, to find the minimum take-off speed, we would need to use the equations for projectile motion and the concept of conservation of energy. This problem highlights the importance of considering all the forces acting on an object and