Multivariable Motion: Marbles Roll off Table, what is speed?

In summary, two marbles with different horizontal speeds roll off a table at the same moment. The faster marble hits the ground first, while the slower marble reaches the ground at the same time due to the effects of gravity. Using the motion of the falling balls, their vertical and horizontal components can be calculated to determine the time of fall and the speed at impact.
  • #1
ixerr
24
0

Homework Statement



Two marbles roll off a table at the same moment. One of the marbles has a horizontal speed of 1 inch per second, and the other has a horizontal speed of 4 inches per second.
a) Which marble hits the ground first?
b) If the table surface is 3 feet from the ground, what are the speeds of the marbles as they hit?

Homework Equations



Don't know, as I can not use normal physics equations to solve for this problem (as my calc professor never mentioned/ used the physics equations to solve such problems in this chapter). Some how derivatives and integrals come into play..

The Attempt at a Solution



a) The marble traveling 4 inches per second hits the ground first. I say this out of common sense though, so I am wondering if there is any mathematical way to figure this out?

b) No idea how to solve for this as we can't use physics equations.
 
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  • #2
Do the marbles start from the same distance to the edge of the table?

ehild
 
  • #3
Yes they do
 
  • #4
ixerr said:
Yes they do

Are you claiming that you are not ALLOWED to use standard physical laws? If so, what *are* you allowed to use? (It is a shame that your "solution" violates several hundred years worth of physical experience.)

RGV
 
  • #5
The question: what is meant with the sentence that the marbles roll off the table at the same moment. If they start rolling on the table from the same distance to the edge, the fastest reaches the edge and starts to fall earlier than the other one. In this case the faster one reaches the ground first. But they both fall for the same time. If they really roll off at the same time, they reach the ground at the same time again.
The motion of the falling balls can be decomposed into a horizontal and vertical component. In the horizontal direction, the ball moves with constant velocity: dx/dt=c (constant). In the vertical direction, the ball moves downwards with constant acceleration: d2y/dt2=g=9.8 m/s2. You see, the vertical motion is not connected to the horizontal velocity. Find dy/dt and y in terms of t. You know that the change of y is 3 feet. Find the time of fall. From that, calculate the vertical velocity when hitting the ground. The vertical and horizontal velocities add up as vectors: the speed is the magnitude of the velocity vector.

ehild
 
Last edited:
  • #6
My interpretation of the phrase "roll off a table at the same moment" would be that they become airborne at exactly the same time.

RGV
 
  • #7
I think the same, but I tried to find a situation when the faster marble reaches the ground first.

ehild
 

1. How is speed defined in multivariable motion?

In multivariable motion, speed is defined as the rate at which an object moves in a particular direction. It is typically measured in units of distance per unit time, such as meters per second or miles per hour.

2. What factors affect the speed of a marble rolling off a table?

The speed of a marble rolling off a table can be affected by several factors, including the height of the table, the angle at which the marble is released, the mass of the marble, and the surface it is rolling on. Air resistance and friction also play a role in determining the speed of the marble.

3. How does the speed of a marble change as it rolls off a table?

The speed of a marble rolling off a table changes due to the influence of various forces acting on it. Initially, the marble will have a high speed as it accelerates downward due to the force of gravity. As it reaches the edge of the table, its speed will decrease due to air resistance and friction. Once it leaves the table and falls to the ground, its speed will increase again due to the force of gravity.

4. Can the speed of a marble rolling off a table be calculated?

Yes, the speed of a marble rolling off a table can be calculated using the equations of motion. These equations take into account the initial speed of the marble, the acceleration due to gravity, and other factors such as air resistance and friction. By plugging in the appropriate values, the speed of the marble at any given point can be determined.

5. How can the speed of a marble rolling off a table be measured?

The speed of a marble rolling off a table can be measured using various instruments and techniques. One method is to use a high-speed camera to record the motion of the marble and then analyze the footage to determine its speed. Alternatively, sensors such as an accelerometer or a photogate can be used to measure the marble's speed directly as it rolls off the table.

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