Velocity, acceleration and speed in space practice problems
Velocity, acceleration and speed of a moving particle, and position from acceleration. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
A particle has position \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(2 t \right)}, \sin{\left(2 t \right)}, t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 0 \).
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle \cos{\left(t \right)}, - \sin{\left(t \right)}, 1 \right\rangle \), initial velocity \( \displaystyle \left\langle -1, 0, -2 \right\rangle \) and initial position \( \displaystyle \left\langle 1, 2, -2 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle \cos{\left(t \right)}, - \sin{\left(t \right)}, 1 \right\rangle \), initial velocity \( \displaystyle \left\langle 3, 1, -5 \right\rangle \) and initial position \( \displaystyle \left\langle -2, -1, 3 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
A particle has position \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 1 \).
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 0, 0, -32 \right\rangle \), initial velocity \( \displaystyle \left\langle -5, 3, 3 \right\rangle \) and initial position \( \displaystyle \left\langle 0, -3, -1 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
A particle has position \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(3 t \right)}, \sin{\left(3 t \right)}, t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = \frac{\pi}{2} \).
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 0, 0, -32 \right\rangle \), initial velocity \( \displaystyle \left\langle 5, -1, -3 \right\rangle \) and initial position \( \displaystyle \left\langle 1, -3, 0 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
A particle has position \( \displaystyle \mathbf r(t) = \left\langle 3 t, 4 t^{2}, 2 t^{3} \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 2 \).
A particle has acceleration \( \displaystyle \mathbf a(t) = \left\langle 0, 0, -32 \right\rangle \), initial velocity \( \displaystyle \left\langle -3, 5, -2 \right\rangle \) and initial position \( \displaystyle \left\langle -2, -2, -2 \right\rangle \). Find \( \displaystyle \mathbf r(t) \).
A particle has position \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \). Find its velocity, acceleration and speed at \( \displaystyle t = 0 \).