traveling on a highway

Question

Suppose you are traveling on a highway at 35 m/s west. Seven seconds later you are traveling 20 m/s northwest. What the magnitude of your acceleration in m/s² to the nearest tenth? What was the direction of your acceleration to the nearest tenth?

Answer 1

Given v(0) = -35i m/s, and v(7)=20 m/s heading NW. Now if we look at a cartesian coordinate system, NW is in the second quadrant where horizontal components are negative and vertical components are positive and NW runs along the angle 90°+45°=135° degrees. That means

v (7) = 20(cos135i + sin135j) = 20(-0.707i + .707j) = -14.1i + 14.1j m/s

To find the average velocity V, we need to add v(0) and v(7):

V = v(0) + v(7) = (-14.1+35)I + 14.1j = 20.9i + 14.1j

The average acceleration A is the average velocity over the elapsed time Δt = 7 sec:

A = V/Δt = (20.9i + 14.1j) m/sec/7 sec = 3i + 2j m/sec²

so │A│= (3² + 2²)^1/2 = 3.6 m/sec²

and the angle of A is tan^-1(2/3) = 33.7°

Solution: 3.6 m/sec² at 33.7°.