You're sailboarding at 7.0 m/s when a wind gust hits, lasting 6.8 s accelerating your board at 0.47 m/s2 at a 35 degrees to your original direction.
#1 and #4 in Related Questions have the formulas.
To find the final velocity and displacement after the wind gust, we can break down the problem into two components: the horizontal component (x-direction) and the vertical component (y-direction).
1. Horizontal Component:
The initial horizontal velocity is 7.0 m/s. The wind gust accelerates the board at 0.47 m/s^2 in a direction 35 degrees to the original direction of motion. To find the horizontal component of acceleration, we use trigonometry:
a_horizontal = acceleration * cos(angle)
a_horizontal = 0.47 m/s^2 * cos(35 degrees)
2. Vertical Component:
The vertical component is unaffected by the wind gust since the gust is perpendicular to the vertical direction. Therefore, the vertical velocity remains constant.
3. Final Velocity:
To find the final velocity, we can calculate the horizontal and vertical components separately and then combine them.
Horizontal Component:
v_horizontal = initial velocity + a_horizontal * time
v_horizontal = 7.0 m/s + (0.47 m/s^2 * cos(35 degrees)) * 6.8 s
Vertical Component:
v_vertical = initial velocity
v_vertical = 7.0 m/s
Final Velocity:
To combine the horizontal and vertical components, we use the Pythagorean theorem:
final velocity = sqrt(v_horizontal^2 + v_vertical^2)
4. Displacement:
To find the displacement, we need to calculate the horizontal and vertical displacements separately:
Horizontal Displacement:
s_horizontal = initial velocity * time + 0.5 * a_horizontal * time^2
s_horizontal = (7.0 m/s * cos(35 degrees)) * 6.8 s + 0.5 * (0.47 m/s^2 * cos(35 degrees)) * (6.8 s)^2
Vertical Displacement:
s_vertical = initial velocity * time
s_vertical = (7.0 m/s) * 6.8 s
Displacement:
To find the total displacement, we use trigonometry and the Pythagorean theorem:
displacement = sqrt(s_horizontal^2 + s_vertical^2)
By plugging in the values and performing the calculations, we can determine the final velocity and displacement after the wind gust.