Question

If a ball is thrown into the air with a velocity of \(40{ft}/{s}\;\), its height in feet tseconds later is given by \(y=40t-16{{t}^{2}}\).

(a) Find the average velocity for the time period beginning when t= 2 and lasting

(i) 0.5 second

(ii) 0.1 second

(iii) 0.05 second

(iv) 0.01 second

(b) Estimate the instantaneous velocity when t = 2.

Short answer

(a) The average velocity for 0.5 seconds, 0.1 seconds, 0.05 seconds, and 0.01 seconds are \(-32{\text{m}}/{\text{s}}\;,-25.6{\text{m}}/{\text{s}}\;,-24.8{\text{m}}/{\text{s}}\;,-24.16{\text{m}}/{\text{s}}\;\) respectively.

(b) The instantaneous velocity at t = 2 is\(-24{\text{m}}/{\text{s}}\;\).

Step-by-step solution

Introduction

The average velocity of any object or material,

\(\text{Avg}\text{. vel}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}}\)

The instantaneous velocity of the object or material,

\(\text{Ins}\text{. vel}=\frac{dy}{dt}\)

Given

The height is given by the equation \(y=40t-16{{t}^{2}}\)

Beginning time t = 2

Velocity \(v=40{ft}/{s}\;\)

Step 3: (a) (i) Average velocity at 0.5 second

Here,\({{t}_{1}}=2,{{t}_{2}}=2.5\)

\(\begin{align}& \text{Avg}\text{. vel}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}} \\ & =\frac{\left( 40\times 2.5-16\times {{2.5}^{2}} \right)-\left( 40\times 2-16\times {{2}^{2}} \right)}{0.5} \\ & =-32{\text{m}}/{\text{s}}\; \end{align}\)

Step 3: (a)(ii) Average velocity at 0.1 second

Here,\({{t}_{1}}=2,{{t}_{2}}=2.1\)

\(\begin{align} & \text{Avg}\text{. vel}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}} \\ &=\frac{\left( 40\times 2.1-16\times {{2.1}^{2}} \right)-\left( 40\times 2-16\times {{2}^{2}} \right)}{0.1} \\ & =-25.6{\text{m}}/{\text{s}}\; \end{align}\)

(a)(iii)Average velocity at 0.05 second 

Here,\({{t}_{1}}=2,{{t}_{2}}=2.05\)

\(\begin{align} & \text{Avg}\text{. vel}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}} \\ & =\frac{\left( 40\times 2.05-16\times {{2.05}^{2}} \right)-\left( 40\times 2-16\times {{2}^{2}} \right)}{0.05} \\ & =-24.8{\text{m}}/{\text{s}}\; \end{align}\)

(a)(iv)Average velocity at 0.01 second

Here,\({{t}_{1}}=2,{{t}_{2}}=2.01\)

\(\begin{align} & \text{Avg}\text{. vel}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}} \\ & =\frac{\left( 40\times 2.01-16\times {{2.01}^{2}} \right)-\left( 40\times 2-16\times {{2}^{2}} \right)}{0.01} \\ & =-24.16{\text{m}}/{\text{s}}\; \end{align}\)

 Instantaneous velocity at t = 2.

\(\begin{align} & \text{Instantaneous velocity}=\frac{d\left( 40t-16{{t}^{2}} \right)}{dt} \\ & =40-32t \end{align}\)

Now at t=2,

\(\begin{align} & \text{Instantaneous velocity}=40-32\times 2\\&=-24{\text{m}}/{\text{s}}\; \end{align}\)