Question

A 55 kg astronaut who weighs 170 N on a distant planet is pondering whether she can leap over a 4.0-m-wide chasm without falling in.

If she leaps at a 15∘ angle, what initial speed does she need to clear the chasm? v=?

Answer 1

Mass of the astronaut, m = 55 kg

Weight of the astronaut at the planet, W = m*g = 170 N

So, g = 170/55 = 3.1 m/s^2

Means, acceleration due to gravity at the planet, g = 3.1 m/s^2

Now, suppose ‘t’ is the total flight time of the astronaut.

Width of the chasm, x = 4.0 m

Suppose, initial speed of the astronaut = v

So, horizontal component of the speed, vx = v*cos15 = 0.97v

Vertical component of the speed, vy = v*sin15 = 0.26v

Now, we have –

vx * t = x

=> t = x / vx = 4.0 / 0.97v = 4.12/v-----------------------------------------(i)

Again, vertical displacement of the astronaut after time t = 0

So –

0 = vy*t – (1/2)*g*t^2

=> 0 = 0.26v*t – 0.5*3.1*t^2

=> 0 = 0.26v*t – 1.55*t^2

Put the value of t from (i) –

0 = 0.26v*(4.12/v) – 1.55*(4.12/v)^2

=> 0 = 1.07 – 26.31 / v^2

=> 26.31 / v^2 = 1.07

=> v^2 = 26.31/1.07

=> v = 4.96 m/s

Therefore, the initial speed of the astronaut, v = 4.96 m/s