A 4 kg mass moves under the influence of a force F⃗=(4t³î-3tĵ)N where t is the time in second. If mass starts from origin at t=0, the velocity and position after t=2 s will be:

Solution

Physics sir · 3 steps

Divide the force by the mass to get acceleration, then integrate twice, one component at a time.

1. Acceleration. Divide the force by m = 4 kg: a⃗ = F⃗/m = t³ î - 3t/4 ĵ. The mass starts from rest at the origin, so v⃗ = 0 and r⃗ = 0 at t = 0.

   v⃗ = ∫₀ᵗ a⃗ dt = t⁴/4 î - 3t²/8 ĵ, r⃗ = ∫₀ᵗ v⃗ dt = t⁵/20 î - t³/8 ĵ

2. Put in numbers. At t = 2 s, velocity: 16/4 = 4 and (3 × 4)/8 = 3/2, so v⃗ = 4 î - 3/2 ĵ m s⁻¹. Position: 32/20 = 8/5 and 8/8 = 1, so r⃗ = 8/5 î - ĵ m.
3. So. v⃗ = 4 î - 3/2 ĵ and r⃗ = 8/5 î - ĵ, which is option (c).

The idea to keep. When force depends on time, find a⃗ = F⃗/m and integrate each component separately, using the starting conditions to fix the constants.

Answer from the NTA final answer key. Solution by PrepWiser.