Find the velocity, acceleration, and speed of a particle with the given position function. r(t) = 5 cos(t)i + 4 sin(t)j

The velocity is the derivative of position.

... v(t) = r'(t) = -5sin(t)i +4cos(t)j . . . . velocity

The acceleration is the derivative of velocity.

... a(t) = v'(t) = -5cos(t)i -4sin(t)j . . . . acceleration

The speed is the magnitude of velocity.

... s(t) = ||v(t|| = √((-5sin(t))² +(4cos(t))²)

... = √(25sin(t)² +16(1 -sin(t)²))

... s(t) = √(16 +9sin(t)²) . . . . speed

Of course, derivatives of these functions are found from ...

... (d/dt)(sin(t)) = cos(t); (d/dt)(cos(t)) = -sin(t); (d/dt)(a + b) = da/dt + db/dt.

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