A speedboat moves on a lake with initial velocity vector v1,x = 8.57 m/s and v1,y = -2.61 m/s, then accelerates for 6.67 seconds at an average acceleration of aav,x = -0.105 m/s2 and aav,y = 0.101 m/s2. What are the components of the speedboat\'s final velocity, v2,x and v2,y?
Find the speedboat's final speed?
Well, let's break it down step by step and calculate the final velocity components.
First, let's find the change in velocity due to acceleration.
Δv = aav * Δt
For x-component:
Δv_x = -0.105 m/s² * 6.67 s = -0.700 m/s
For y-component:
Δv_y = 0.101 m/s² * 6.67 s = 0.674 m/s
Now, let's add the change in velocity to the initial velocity to find the final velocity.
For x-component:
v2,x = v1,x + Δv_x = 8.57 m/s - 0.700 m/s = 7.87 m/s
For y-component:
v2,y = v1,y + Δv_y = -2.61 m/s + 0.674 m/s = -1.94 m/s
So, the speedboat's final velocity components are v2,x = 7.87 m/s and v2,y = -1.94 m/s.
To find the speedboat's final speed, we can use the Pythagorean theorem:
v2 = sqrt((v2,x)² + (v2,y)²)
Calculating that, we get:
v2 = sqrt((7.87 m/s)² + (-1.94 m/s)²) ≈ 7.98 m/s
Therefore, the speedboat's final speed is approximately 7.98 m/s. But hey, don't worry – boats are used to speed up and slow down, whereas clowns are just used to confuse you!
To find the components of the speedboat's final velocity, v2,x and v2,y, we can use the following kinematic equation:
v2 = v1 + aav * t
where v1 is the initial velocity, aav is the average acceleration, and t is the time.
Given:
v1,x = 8.57 m/s
v1,y = -2.61 m/s
aav,x = -0.105 m/s^2
aav,y = 0.101 m/s^2
t = 6.67 s
First, let's find v2,x:
v2,x = v1,x + aav,x * t
v2,x = 8.57 m/s + (-0.105 m/s^2) * 6.67 s
v2,x = 8.57 m/s - 0.7 m/s
v2,x = 7.87 m/s
Next, let's find v2,y:
v2,y = v1,y + aav,y * t
v2,y = -2.61 m/s + 0.101 m/s^2 * 6.67 s
v2,y = -2.61 m/s + 0.674 m/s
v2,y = -1.93 m/s
Therefore, the components of the speedboat's final velocity are:
v2,x = 7.87 m/s
v2,y = -1.93 m/s
To find the speedboat's final speed, we can use the Pythagorean theorem:
v2 = sqrt(v2,x^2 + v2,y^2)
v2 = sqrt((7.87 m/s)^2 + (-1.93 m/s)^2)
v2 = sqrt(61.9369 + 3.7249) m/s
v2 = sqrt(65.6618) m/s
v2 = 8.11 m/s
Therefore, the speedboat's final speed is approximately 8.11 m/s.
To find the speedboat's final velocity components, we can use the equations of motion.
The equation for the final velocity in terms of initial velocity, acceleration, and time is:
vf = vi + at
Now let's break down the equations for both x and y components:
For the x-component:
vf,x = vi,x + aav,x * t
Given:
vi,x = 8.57 m/s (initial velocity in the x-direction)
aav,x = -0.105 m/s² (average acceleration in the x-direction)
t = 6.67 seconds (time)
Substituting the values, we get:
vf,x = 8.57 m/s + (-0.105 m/s²) * 6.67 s
vf,x = 8.57 m/s - 0.70135 m/s
vf,x = 7.86865 m/s (rounded to four decimal places)
For the y-component:
vf,y = vi,y + aav,y * t
Given:
vi,y = -2.61 m/s (initial velocity in the y-direction)
aav,y = 0.101 m/s² (average acceleration in the y-direction)
t = 6.67 seconds (time)
Substituting the values, we get:
vf,y = -2.61 m/s + (0.101 m/s²) * 6.67 s
vf,y = -2.61 m/s + 0.67467 m/s
vf,y = -1.93533 m/s (rounded to four decimal places)
Therefore, the components of the speedboat's final velocity are:
v2,x = 7.8687 m/s (rounded to four decimal places)
v2,y = -1.9353 m/s (rounded to four decimal places)
To find the speedboat's final speed, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle (c) is equal to the sum of the squares of the other two sides (a and b). In this case, the horizontal and vertical components of the final velocity represent the two sides of the triangle. The hypotenuse represents the final speed (v2).
Using the equation:
v2 = sqrt(v2,x^2 + v2,y^2)
Substituting the values, we get:
v2 = sqrt((7.8687 m/s)^2 + (-1.9353 m/s)^2)
v2 = sqrt(61.90553769 m²/s² + 3.74516009 m²/s²)
v2 = sqrt(65.65069778 m²/s²)
v2 ≈ 8.11 m/s (rounded to two decimal places)
Therefore, the speedboat's final speed is approximately 8.11 m/s.
Xo = 8.57 m/s
Yo = -2.61 m/s
a. X = Xo + a*t = 8.57 + (-0.105)*6.67 = 7.87 m/s
Y = Yo + a*t = -2.61 + 0.101*6.67 = -1.94 m/s
b. V^2 = X^2 + Y^2 = 7.87^2 + 1.94^2 =
65.7
V = 8.11 m/s.