A mountain climber jumps a 2.8 m -wide crevasse by leaping horizontally with a speed of 7.6 m/s .

Question

A)If the climber's direction of motion on landing is -45 ∘, what is the height difference between the two sides of the crevasse?

B)Where does the climber land?

Answer 1

This problem requires a particular property of parabolas.

I do not know your math background, so let me just state it as a lemma with a proof

using calculus. If you did not have calculus, you will just need to believe me.

Lemma: A parabola given by y = kx² has slope s at the coordinates (s/2k, s²/4k).

Proof:

y' = 2kx

The constraint on the slope gives the equation:

2kx = s

Solving for x:

x = s/2k

Plugging the value of x into the equation of the parabola

y = k(s/2k)² = s²/4k

QED

Now for the physics.

Let

v = 7.6 m/s be the horizontal speed of the jump,

g = 9.81 m/s² be gravitational acceleration.

The trajectory of the climber will be a parabola resulting from a combination of two motions:

- A horizontal trajectory, which the climber would follow in the absence of gravity

- A downward fall, which the climber would suffer in the absence of any jump.

You can view the trajectory in an X-Y coordinate space, where

the X coordinate is horizontal, and

the Y coordinate is vertical, with origin being at the start point of the jump.

Then [x(t), y(t)] is the position of the climber after time t.

(We start measuring time t from the instant the climber leaves the surface.)

x(t) = vt

y(t) = -gt²/2,

The first equation gives

t = x(t)/v

Plugging into the second equation

y(t) = -gx²(t)/2v²

The resulting trajectory is a parabola expressed as a function of x:

y = -gx²/2v²

Now we can apply the lemma with

k = -g/2v²

s = -1 (which is the slope of -45°)

The lemma gives us landing coordinates

(v²/g, -v²/2g)

Plugging in actual values,

the climber lands at height

y = -7.6²/(2×9.81) = -2.9 m

and horizontal distance

x = 7.6²/9.81 = 5.8 m

This implies two things

- The climber made it to the other side and landed 3 m from the edge.

- The height difference is 2.9 m.