Solved problem: Acceleration of a meteorite

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Statement

difficulty

A meteorite moves through space with a velocity v⃗(t) = (1+4·t) i⃗+t² j⃗ m. Calculate

a) Its average acceleration between the times t₁=2 s and t₂= 4 s.
b) Its acceleration at t₃= 6 s.

Solution

Question a)

Data

v⃗(t) = (1+4·t) i⃗+t² j⃗ m

t₁=2 s y t₂=4 s

Resolution

To calculate the average acceleration we must use the following equation:

{\vec{a}}_{a} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{t_{2} - t_{1}} = \frac{∆ \vec{v}}{∆ t}

We know t₁ and t₂. Now we need to calculate the velocity at the instant t₁ (v⃗₁) and at the time t₂ (v⃗₂). So, just substitute in the equation of velocity that was provided in the statement of the exercise:

For t₁=2 s

${\vec{v}}_{1} = v (2) = (1 + 4 \cdot 2) \cdot \vec{i} + (2)^{2} \cdot \vec{j} m / s \Rightarrow {\vec{v}}_{1} = 9 \cdot \vec{i} + 4 \cdot \vec{j} m / s$

For t₂=4 s

${\vec{v}}_{2} = \vec{v} (4) = (1 + 4 \cdot 4) \cdot \vec{i} + (4)^{2} \cdot \vec{j} m / s \Rightarrow {\vec{v}}_{2} = 17 \cdot \vec{i} + 16 \cdot \vec{j} m / s$

Substituting in the first equation:

${\vec{a}}_{a} = \frac{(17 - 9) \cdot \vec{i} + (16 - 4) \cdot \vec{j}}{4 - 2} m / s^{2} \Rightarrow {\vec{a}}_{a} = \frac{8 \cdot \vec{i} + 12 \cdot \vec{j}}{2} m / s^{2} \Rightarrow {\vec{a}}_{a} = 4 \cdot \vec{i} + 6 \cdot \vec{j} m / s^{2}$

Question b)

Data

v⃗(t) = (1+4·t) i⃗+t² j⃗ m
t₃= 6 s

Resolution

To calculate the acceleration at time t₃ we must first calculate the average acceleration:

\vec{a} = \lim_{∆ t \to 0} {\vec{a}}_{a} = \lim_{∆ t \to 0} \frac{∆ \vec{v}}{∆ t} = \frac{d \vec{v}}{d t}

Taking the derivative of velocity with respect to time we get that:

$\vec{a (t)} = 4 \cdot \vec{i} + 2 \cdot t \cdot \vec{j} m / s^{2}$

and substituting the value for t₃= 6 s.

$\vec{a (6)} = 4 \cdot \vec{i} + 2 \cdot 6 \cdot \vec{j} m / s^{2} \Rightarrow \vec{a (6)} = 4 \cdot \vec{i} + 12 \cdot \vec{j} m / s^{2}$

Formulas worksheet

These are the main formulas that you must know to solve this exercise. If you are not clear about their meaning, we recommend you to check the theory in the corresponding sections. Furthermore, you will find in them, under the Formulas tab, the codes that will allow you to integrate these equations in external programs like Word or Mathematica.

Formulas

Related sections

{\vec{a}}_{a} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{t_{2} - t_{1}} = \frac{∆ \vec{v}}{∆ t}

Average Acceleration

\vec{a} = \lim_{∆ t \to 0} {\vec{a}}_{a} = \lim_{∆ t \to 0} \frac{∆ \vec{v}}{∆ t} = \frac{d \vec{v}}{d t}

Instantaneous Acceleration

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Solution

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Acceleration of a meteorite

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