# The constant forces F1 = 8i + 12j N and F2 = 4i - 4j N act on a particle with mass 4 kg. If v=40i + 32j when t = 20, show that v = -20i - 8j when t=0. When is the speed of the particle 8 m/s.

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The constant forces F1 = 8i + 12j N and F2 = 4i - 4j N act on a particle with mass 4 kg.

The net force on the particle due to the two forces is F = 12i + 8j. The acceleration as a result of this is a = F/m = 3i + 2j.

If v = 40i + 32j when t = 20, the initial velocity is 40i + 32j - 20(3i + 2j)

=> 40i - 60i + 32j - 40j

=> -20i - 8j m/s

The velocity of the particle at a time t is -20i + 3ti - 8j + 2tj

The speed is `sqrt( (-20 + 3t)^2 + (2t - 8)^2))`

If the speed is 8, `sqrt( (-20 + 3t)^2 + (2t - 8)^2)) = 8`

=> (-20 + 3t)^2 + (2t - 8)^2) = 64

=> 400 + 9t^2 - 120t + 4t^2 + 64 - 32t = 64

=> 13t^2 - 152t + 400 = 0

=> t1 = `100/13` , t2 = 4

**The speed is 8 m/s at 100/13 s and 4 s.**

At time t seconds, the velocity of the particle is vms^-1. When t=20, v=40i+32j. Show that v=-20i-8j when t=0.

Write down an expression for v at time t.