Question

The position of a small airplane relative to an airport is described by a vector equation ? ⃗(t) = x (t) ?̂ + y (t) ?̂ + z (t) ? ̂ with ?̂, ?̂, ? ̂ unit vectors where x(t) = 130 t (km/h) – 50.0 t 2 (km/h2) y(t) = 100 t (km/h) – 25.0 t 2 (km/h2) z(t) = 2.00 t (km/h) – 0.500 t 2 (km/h2).

(a) Calculate the plane’s instantaneous velocity ? ⃗ in km/h at t = 1.00 h.





(b) Calculate the plane’s instantaneous acceleration ? ⃗ in km/h2 at t = 1.00 h.

Answer 1

here,

the position vector , r(t) = (130 t - 50 t^2 ) i + (100 t - 25 t^2) j + (2 t - 0.5 * t^2) j

a)

the plane’s instantaneous velocity , v(t) = dr(t) /dt

v(t) = ( 130 - 100 t) i + (100 - 50 t) j + ( 2 - t) k

at t = 1 h

v(1) = ( 130 - 100 * 1) i + (100 - 50 * 1) j + ( 2 - 1) k

v(1) = (30 i + 50 j + 1 k ) km/h

b)

the plane’s instantaneous acceleration , a(t) = dv(t) /dt

a(t) = ( - 100 ) i + ( - 50) j + (- 1) k

a(t) = (- 100 i - 50 j - k) km/h^2

the acceleration is constant

the acceleration is a(1) = (- 100 i - 50 j - k) km/h^2