Question

After discovering that two hikers have been stranded on Devil's Thumb Pass, the Colorado Search and

Rescue team sends out a plane with supplies for the hikers. Because the hikers are in an isolated location, the

plane has to drop the supplies as it ies overhead. If the plane is 235 m above the hikers and

has a horizontal speed of 200 km/h, how far away from the hikers should the supplies be dropped if they are

to land at the hikers' location? (Hint: the initial y-component of the supplies' velocity is zero

Answer 1

Solution: We consider the horizontal direction as X direction and vertical direction as Y direction. Further we consider the origin at the plane the moment it drops the supplies.

Thus the plane’s height above the hiker h = 235 m

Horizontal speed of the supplies when they were dropped; v = 200 km/h = (200 km/hr)*(1000m/1km)/(3600s/1hr) = 55.56 m/s

The angle made by the supplies when were being dropped θ = 0^o

The horizontal component of the velocity v_x = v*cosθ = 55.56*cos(0^o) = 55.56 m/s

The vertical component of the velocity v_y = v*sinθ = 55.56*sin(0^o) = 0 m/s

Its means that the vertical velocity of the supplies is zero as they were just dropped (not thrown) from a horizontally moving plane. However the supplies have the horizontal velocity equal to 55.56 m/s initially.

When the supplies reach victims, the vertical displacement of the supplies is y = -235 m. (Negative sign taken for the distance fallen by the supplies in the negative Y direction)

The time taken by the supplies to fall this vertical distance is given by,

y = v_y*t +(1/2)*g*t²

-235m = (0m/s)t + (1/2)*(-9.81)* t²

t² = 47.91 s²

t = 6.9217 s.

Thus the supplies fall 235 m in 6.9217 seconds.

During this fall, the supplies also move horizontally with the velocity of v_x = 55.56 m/s.

(The horizontal component of velocity do not change as there is no acceleration acting in the horizontal direction)

Thus the horizontal distance travelled by the supplies in t = 6.9217 s is given by

x = v_x*t

x = (55.56 m/s)*(6.9217 s)

x = 384.57 m.

Thus the supplies should be dropped 384.57 m or 385 m away from the hiker (before flying overhead) if they to land at the hiker’s location.