Landing question

Question

Two kittens are running toward each other, one is named Chloe and the other is named Luna. Luna runs at a speed of 1.40 m/s. If Chloe jumps at an angle of 13.0° with speed 2.00 m/s, at what distance from Luna should Chloe jump to make sure she lands on Luna?

Answer 1

Let

w = 1.4 m/s be Luna's speed,

v = 2 m/s be Chloe's speed,

α = 13° be the angle of Chloe's jump,

s (to be calculated) be Luna's distance from Chloe at the time of jump,

g = 9.81 m/s² be gravitational acceleration.

Imagine that the two cats are moving in an xy-plane,

where the origin is at the point of Chloe's jump,

the x-axis is horizontal and extends from the point of jump towards Luna,

and y is the vertical axis.

Let x_c(t) and y_c(t) be Chloe's coordinates at a time t, and

let x_l(t) and y_l(t) be Luna's coordinates at a time t.

We let time start with the jump, so

x_c(0) = 0

y_c(0) = 0

x_l(0) = s

y_l(0) = 0

At any subsequent time t the coordinates are

x_c(t) = vcos(α)t

y_c(t) = vsin(α)t - gt²/2

x_l(t) = s - wt

y_l(t) = 0

(If you do not know where these equations come from, let me know.)

We are to find s, so that there exists time t₁ when Chloe is on top of Luna,

i.e., their coordinates are the same:

x_c(t₁) = x_l(t₁)

y_c(t₁) = y_l(t₁)

Substituting x_c, y_c, x_l, y_l:

vcos(α)t₁ = s - wt₁ (1)

vsin(α)t₁ - gt₁²/2 = 0 (2)

So we have two equations (1), (2) with two unknowns s and t₁.

We can rewrite equation (1):

s = (vcos(α) + w)t₁ (3)

We can calculate t1 from equation (2). It has two solutions:

t₁ = 0, and t₁ = 2vsin(α)/g

The solution t₁=0 means that at time 0 the two cats have the same y-coordinates,

which is true, but not useful for us.

The second solution is useful, and we can substitute it into (3):

s = (vcos(α) + w)2vsin(α)/g

Substituting actual numbers

s = (1.4×cos(13°) + 2)×2×1.4×sin(13°)/9.81 = 0.216

So Chloe should jump when Luna is about 21.6 cm away.

Please note that this answer makes no sense in reality, as 21 cm is about the length of a cat.

To solve this problem we had to treat both cats as if they were mere dots.