How Do You Calculate the Time a Marble Takes to Travel from L to P?

[BraindeadX64]

Homework Statement

A very small steel marble is shown rolling at a constant speed on a horizontal table. The marble leaves the table at M, falls, and hits the ground at P. This is illustrated in the diagram below which is drawn to scale. Calculate the time it took the marble to travel from L to P.

Homework Equations

Professor Posted This

Still don't understand

p(t) = 0.5 * a * t^2 + v0 * t + p0
where
p(t) is position at time t
a is acceleration
v0 is initial velocity at time 0
p0 is position at time 0

The Attempt at a Solution

Vertical acceleration due to gravity = -9.8 m/s^2 = -980 cm/s^2

px(t) = 100 = 0.5 * 0 * t^2 + v0 * t + 60 = v0 * t + 60
vertical
py(t) = 0 = 0.5 * -980 * t^2 + 0 * t + 40 = -490 * t^2 + 40

t=2.85s

v0 = 210.011

got an answer and online submition said it was wrong

Please help

 

[gomunkul51]

Basically you need to find how long it took for the ball to fall 70cm, then you will know the horizontal velocity.

x = x₀ + v₀t + (1/2)at²
40 = 0 + 0 + (1/2)gt²
t = sqrt(80/980) ~ 0.286 sec // you wrote t=2.85s, missed a zero ;)

then v horizontal is:

x = x₀ + v₀t + (1/2)at²

100 = 40 + v_ht + 0

v_h = 60/sqrt(80/980) = 210 cm/sec


** The question your Proff. explained has the same idea but different numbers.

 

[verty]

There are perhaps two ways to do a question like this. One is to use physical insight to see the essence of the problem so you don't have to think too much about what signs values should be and so on. Another way is to leave nothing to chance and do everything the way it is given.

If you do the first, you would consider that you need the total time, which is just the sum of the two times from L to M and M to P. Each can be done independently, just to be added together afterwards.

Time L to M can't be done just yet because you don't know the speed of the marble. So start with time M to P. That is just the time it took the marble to fall 40cm.

Once you have that, you need to find the speed of the marble, which you can find using the time from M to P.

So you see, this is decomposing the problem into managable chunks, then doing each easy piece.


The other way is as follows but uses less physical insight. I think it is less nice:

At time t = 0, the marble is at L.
At time t = t_m, the marble is at M.
At time t = t_p, the marble is at P.

At time t = t_m:

s_x = 20 + v_x t_m = 40
s_y = 40

At time t = t_p:

s_x = 40 + v_x (t_p - t_m) = 100
s_y = 40 - (1/2) g (t_p - t_m)^2 = 0

We have three equations in three unknowns, v_x, t_m, t_p:

20 + v_x t_m = 40
40 + v_x (t_p - t_m) = 100
40 - (1/2) g (t_p - t_m)^2 = 0

Solve simultaneously to find t_p.