Question

Find the velocity v(t) and speed ∥v(t)∥ of a particle whose motion is described by x=2,y=8t³−60t²,z=t²−10t+25 v(t)=

∥v(t)∥=
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Answer 1

To find the velocity vector v(t) and the speed ||v(t)|| of a particle whose motion is described by the parametric equations:

x(t) = 2

y(t) = 8t^3 - 60t^2

z(t) = t^2 - 10t + 25

We'll need to find the derivatives of x(t), y(t), and z(t) with respect to time t, which will give us the components of the velocity vector v(t). Then, we can find the magnitude of the velocity vector to determine the speed ||v(t)||.

• Derivatives of x(t), y(t), and z(t) with respect to t:

x'(t) = d(2)/dt = 0 (since x(t) is a constant)

y'(t) = d/dt(8t^3 - 60t^2) = 24t^2 - 120t

z'(t) = d/dt(t^2 - 10t + 25) = 2t - 10

So, the components of the velocity vector v(t) are:

v_x(t) = 0

v_y(t) = 24t^2 - 120t

v_z(t) = 2t - 10

• To find the magnitude of the velocity vector ||v(t)||, we use the formula:

||v(t)|| = sqrt((v_x(t))^2 + (v_y(t))^2 + (v_z(t))^2)

Substitute the values of v_x(t), v_y(t), and v_z(t) into the formula:

||v(t)|| = sqrt((0)^2 + (24t^2 - 120t)^2 + (2t - 10)^2)

Simplify this expression:

||v(t)|| = sqrt((24t^2 - 120t)^2 + (2t - 10)^2)

This is the magnitude of the velocity vector at any given time t.

Keep in mind that this is a general expression for ||v(t)||. If you need to find the speed at a specific time t, you would substitute the value of t into this expression.

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Answer 2

To find the velocity vector v(t) and the speed ∥v(t)∥ of the particle, we can differentiate the position vector x(t) with respect to time.

Given:

x = 2

y = 8t³ - 60t²

z = t² - 10t + 25

Differentiating each component with respect to time, we have:

dx/dt = 0

dy/dt = 24t² - 120t

dz/dt = 2t - 10

Thus, the velocity vector v(t) = (dx/dt, dy/dt, dz/dt) becomes:

v(t) = (0, 24t² - 120t, 2t - 10)

To find the speed ∥v(t)∥, we need to calculate the magnitude of the velocity vector v(t).

The magnitude of a vector can be calculated using the formula:

∥v(t)∥ = √(x² + y² + z²)

Substituting the components of v(t) into the formula, we have:

∥v(t)∥ = √((0)² + (24t² - 120t)² + (2t - 10)²)

∥v(t)∥ = √(0 + (24t² - 120t)² + (2t - 10)²)

Simplifying the expression further, we have:

∥v(t)∥ = √((24t² - 120t)² + (2t - 10)²)

∥v(t)∥ = √(576t^4 - 5760t^3 + 24000t^2 + 4t^2 - 40t + 100)

Finally, we can simplify the expression by combining like terms and taking the square root to find the speed ∥v(t)∥.

Please note that if you need to find the specific velocity and speed at a certain time t, you can substitute the value of t into the expressions for v(t) and ∥v(t)∥.