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A hug is mowing hack and forth on a straight path. The velocity of the bug is given by v(t)=t^(2)-3t. Find the average acceleration of the bug on the interval [1,4].

A hug is mowing hack and forth on a straight path. The velocity of the bug is given by v(t)=t2−3t v(t)=t^{2}-3 t . Find the average acceleration of the bug on the interval [1,4] [1,4] .

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Relate position, velocity, speed, and acceleration using derivativesright chevron icon

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Q. A hug is mowing hack and forth on a straight path. The velocity of the bug is given by v(t)=t2−3t v(t)=t^{2}-3 t . Find the average acceleration of the bug on the interval [1,4] [1,4] .
  1. Calculate Average Acceleration: To find the average acceleration, we need to calculate the change in velocity Δv\Delta v over the change in time Δt\Delta t.
  2. Find Velocity at t=1t=1: First, find the velocity at t=1t=1 using the velocity function v(t)=t2−3tv(t)=t^2-3t.v(1)=(1)2−3(1)=1−3=−2v(1) = (1)^2 - 3(1) = 1 - 3 = -2.
  3. Find Velocity at t=4t=4: Now, find the velocity at t=4t=4 using the same function.v(4)=(4)2−3(4)=16−12=4.v(4) = (4)^2 - 3(4) = 16 - 12 = 4.
  4. Calculate Change in Velocity: Calculate the change in velocity, Δv\Delta v, which is v(4)−v(1)v(4) - v(1). Δv=4−(−2)=4+2=6\Delta v = 4 - (-2) = 4 + 2 = 6.
  5. Calculate Change in Time: Calculate the change in time, Δt\Delta t, which is 4−14 - 1.\newlineΔt=4−1=3\Delta t = 4 - 1 = 3.
  6. Calculate Average Acceleration: Now, calculate the average acceleration, aavga_{\text{avg}}, which is Δv/Δt\Delta v/\Delta t.\newlineaavg=63=2a_{\text{avg}} = \frac{6}{3} = 2.

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