Question

please explain and draw out picture if apllicable

Zeta Anderson, futuristic super-spy for the Terran Confederation needs to reach her stealth vessel 100m away, or risk either discovery, or running out of air in deep space. For our purposes, there are two relevant directions, the direction Zeta is traveling in (the x-direction, at 1 m/s), and direction of the stealth ship's motion, which is perpendicular to her motion (y-direction, 10 m/s). This is strange as she left orders for the ship to remain stationary with respect to the Zorn ship. Zeta uses her jet-pack, choosing to accelerate directly in the x-direction at 0.25g (she starts at (x=100,y=0)). If the stealth vessel is 25 m long, and its nose cone starts at x=0,y=0, will Zeta reach the ship or miss? If she misses, how should she have chosen her path (assume she cannot exceed 1 g)?

Answer 1

Figure 1

We are given that Zeta was travelling initially with a velocity of 1m/s in the x direction, then she decided to accelerate with 2.5g in the x direction when the nose cone of her ship is at (0,0) co-ordinate

We can see that Zeta has to travel 100m along the negative x direction before her ship reaches above the x axis as shown in the figure

Let us calculate how much time will it require for the ship to reach above the x axis, ie the nose cone should displace from (0,0) to (0,25), since the ship is 25m long. It is travelling at 10m/s in the y direction, hence

Time taken,

Hence, in order to reach her ship, Zeta must travel a distance of 100m in 2.5s

Now let's find out how much distance can she travel in 2.5s, for that purpose let us use the equation of motion,

Where s is the distance covered in t seconds, u is the initial velocity=1m/s, t=2.5s

Substituting everthing in the equation:

Which means, she will certainly not reach her ship in time.

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Its pretty obvious that she should travel at some angle away from the x axis in order to reach her ship in time:

Figure 2

Let us assume that Zeta decided to wisen up and travel at an angle from the x axis and at an acceleration of 1g in that direction as shown in the figure, then

                ..................(1)

We can see that metres, where t is the time at which Zeta supposedly reaches her ship

Sine Zeta should reach her hip at this time, the distance traveled by her vertical component of the acceleration must be , therefore:

, (since she has no initial velocity in y direction)

ie,

which implies,

Now, let us move on to find :

We know that,           ...............(2)

The distance travelled by Zeta in x direction will be given by,

ie, ( = 1m/s)

Which implies,

Let us substitute the above expression into equation(2):

ie,

On solving the above solution we can obtain (taking only the positive solution)

Saubtituting the value of t back in equation (1):

Therefore, she has to travel with an acceleration of 1g at an angle of from the x direction