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Given the displacement S = t4 + 3t2 + 4 metres.

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Here's the solution for question 5:

Given the displacement S=t4+3t2+4S = t^4 + 3t^2 + 4 metres.

Step 1: Find the general expression for velocity.
Velocity vv is the first derivative of displacement with respect to time tt.

v=dSdt=ddt(t4+3t2+4)v = \frac{dS}{dt} = \frac{d}{dt}(t^4 + 3t^2 + 4) v=4t3+6t m/sv = 4t^3 + 6t \text{ m/s}

Step 2: Find the general expression for acceleration.
Acceleration aa is the first derivative of velocity with respect to time tt.

a=dvdt=ddt(4t3+6t)a = \frac{dv}{dt} = \frac{d}{dt}(4t^3 + 6t) a=12t2+6 m/s2a = 12t^2 + 6 \text{ m/s}^2

Part a· 1 step

At t=0t=0 s

  1. Calculate displacement, velocity, and acceleration at t=0t=0.

    • Displacement:
    S(0)=(0)4+3(0)2+4=4 mS(0) = (0)^4 + 3(0)^2 + 4 = 4 \text{ m}
    • Velocity:
    v(0)=4(0)3+6(0)=0 m/sv(0) = 4(0)^3 + 6(0) = 0 \text{ m/s}
    • Acceleration:
    a(0)=12(0)2+6=6 m/s2a(0) = 12(0)^2 + 6 = 6 \text{ m/s}^2

    At t=0t=0: Displacement = 4 m\boxed{\text{4 m}}, Velocity = 0 m/s\boxed{\text{0 m/s}}, Acceleration = 6 m/s2\boxed{\text{6 m/s}^2}.

Part b· 1 step

At t=1t=1 s

  1. Calculate displacement, velocity, and acceleration at t=1t=1.

    • Displacement:
    S(1)=(1)4+3(1)2+4=1+3+4=8 mS(1) = (1)^4 + 3(1)^2 + 4 = 1 + 3 + 4 = 8 \text{ m}
    • Velocity:
    v(1)=4(1)3+6(1)=4+6=10 m/sv(1) = 4(1)^3 + 6(1) = 4 + 6 = 10 \text{ m/s}
    • Acceleration:
    a(1)=12(1)2+6=12+6=18 m/s2a(1) = 12(1)^2 + 6 = 12 + 6 = 18 \text{ m/s}^2

    At t=1t=1: Displacement = 8 m\boxed{\text{8 m}}, Velocity = 10 m/s\boxed{\text{10 m/s}}, Acceleration = 18 m/s2\boxed{\text{18 m/s}^2}.

Part c· 1 step

At t=4t=4 s

  1. Calculate displacement, velocity, and acceleration at t=4t=4.

    • Displacement:
    S(4)=(4)4+3(4)2+4=256+3(16)+4=256+48+4=308 mS(4) = (4)^4 + 3(4)^2 + 4 = 256 + 3(16) + 4 = 256 + 48 + 4 = 308 \text{ m}
    • Velocity:
    v(4)=4(4)3+6(4)=4(64)+24=256+24=280 m/sv(4) = 4(4)^3 + 6(4) = 4(64) + 24 = 256 + 24 = 280 \text{ m/s}
    • Acceleration:
    a(4)=12(4)2+6=12(16)+6=192+6=198 m/s2a(4) = 12(4)^2 + 6 = 12(16) + 6 = 192 + 6 = 198 \text{ m/s}^2

    At t=4t=4: Displacement = 308 m\boxed{\text{308 m}}, Velocity = 280 m/s\boxed{\text{280 m/s}}, Acceleration = 198 m/s2\boxed{\text{198 m/s}^2}.

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