Ezekiel U. answered • 05/04/23
Honors engineering student with 3+ years expertise teaching physics
The answer is 4.63 meters.
We're being asked to find the egg's change in horizontal distance, Δx. The initial velocity is the sum of the initial velocity in the x direction and the initial velocity in the y direction, which we can find by representing the original velocity vector as the hypotenuse of a triangle with the y component being the vertical leg and the x competent being the horizontal leg. Therefore,
v0 = vx + vy
vx = v0 · cos(20°), and vy = v0 · sin(20°)
Where v0 is the initial velocity. Recall now that velocity times time equals displacement, so vx · Δt = Δx. We know vx = v · cos(20°) = 8.4 · cos(20°) m/s, but we don't yet know Δt. We can find it by thinking about what's happening in the y direction. Let's use the kinematic equation Δd = v0 · t + (1/2) · a · t2. Since the egg starts at ground level and hits the ground at the end of its flight, Δd = 0. a will be the acceleration due to gravity, since we assume no other force is acting on the egg. We then have
0 = v0 · t + (1/2) · (-9.8 m/s2) · t2
Which, if we rearrange, gives us
t = -2 · v0 / (-9.8 m/s2)
and since we're considering motion in the y direction, v0 = vy = v0 · sin(20°). That means the time is
t = -2 · v0 · sin(20°) / (-9.8 m/s2) = 0.586 seconds.
The reason we cared about time is so we could plug it back into the equation vx · Δt = Δx. Doing so now gives us
Δx = 0.586 s · 8.4 m/s · cos(20°) = 4.63 m, our final answer.
