A golfer hits a golf ball at an angle of with initial velocity .
(a) Estimate if the ball is hit the same velocity but the angle increased by .
(b) Estimate if the ball is hit the same angle but the velocity increased by 4 .
(Give your answer to four decimal places.)
(a)
(b)

Ask by Morrison Wilson. in the United States

Mar 22,2025

Question

A golfer hits a golf ball at an angle of with initial velocity .
(a) Estimate if the ball is hit the same velocity but the angle increased by .
(b) Estimate if the ball is hit the same angle but the velocity increased by 4 .
(Give your answer to four decimal places.)
(a)
(b)

Ask by Morrison Wilson. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We start with the formula for the range of a projectile (ignoring air resistance)

$$
s=\frac{v^2\sin(2\theta)}{g},
$$

where

• $$ v $$ is the initial speed,  
• $$ \theta $$ is the launch angle, and  
• $$ g $$ is the acceleration due to gravity (we use $$ g=32.2\; \mathrm{ft/s^2} $$).

We use the method of linear approximation. A small change in $$ s $$ from a change in $$ \theta $$ or $$ v $$ is given (keeping the other variable fixed) by

$$
\Delta s\approx\frac{\partial s}{\partial \theta}\Delta\theta \quad \text{or} \quad \Delta s\approx\frac{\partial s}{\partial v}\Delta v,
$$

respectively.

---

### (a) Change in Angle by $$ 3^\circ $$

The partial derivative of $$ s $$ with respect to $$ \theta $$ is

$$
\frac{\partial s}{\partial \theta}=\frac{v^2\cdot 2\cos(2\theta)}{g}.
$$

At our given values, $$ v=110\; \mathrm{ft/s} $$ and $$ \theta=24^\circ $$, we have $$ 2\theta=48^\circ $$. Thus

$$
\frac{\partial s}{\partial \theta}=\frac{110^2\cdot 2\cos(48^\circ)}{32.2}.
$$

Calculating step by step:

1. $$ 110^2=12100 $$.
2. $$ \cos(48^\circ)\approx0.6691 $$.
3. Then, 
   $$
   \frac{\partial s}{\partial \theta}\approx\frac{12100\cdot2\cdot0.6691}{32.2}=\frac{12100\cdot1.3382}{32.2}.
   $$
4. Multiply the numerator:
   $$
   12100\cdot1.3382\approx16195.
   $$
5. Divide by $$ 32.2 $$:
   $$
   \frac{16195}{32.2}\approx503.1.
   $$

Since the change in angle is $$ 3^\circ $$ we must convert it to radians:

$$
\Delta\theta=3^\circ\times\frac{\pi}{180}=\frac{\pi}{60}\approx0.05236\, \text{radians}.
$$

Thus the change in distance is approximately

$$
\Delta s\approx503.1\times\frac{\pi}{60}\approx503.1\times0.05236\approx26.3280\; \mathrm{ft}.
$$

---

### (b) Change in Velocity by $$ 4 \; \mathrm{ft/s} $$

Next, we take the partial derivative of $$ s $$ with respect to $$ v $$:

$$
\frac{\partial s}{\partial v}=\frac{2v\sin(2\theta)}{g}.
$$

At $$ v=110\; \mathrm{ft/s} $$ and $$ 2\theta=48^\circ $$ (with $$\sin(48^\circ)\approx0.7431$$), we have

$$
\frac{\partial s}{\partial v}=\frac{2\cdot110\cdot0.7431}{32.2}.
$$

Step by step:

1. $$ 2\cdot 110=220 $$.
2. $$ 220\cdot0.7431\approx163.482 $$.
3. Then,
   $$
   \frac{\partial s}{\partial v}\approx\frac{163.482}{32.2}\approx5.077.
   $$

For an increase of $$ \Delta v=4\; \mathrm{ft/s} $$:

$$
\Delta s\approx5.077\times4\approx20.3080\; \mathrm{ft}.
$$

(To four decimal places, $$20.3080$$ ft.)

---

### Final Answers

(a) $$\Delta s\approx26.3280\; \mathrm{ft}$$  
(b) $$\Delta s\approx20.3080\; \mathrm{ft}$$
(a) Δs ≈ 26.3280 ft (b) Δs ≈ 20.3080 ft

A golfer hits a golf ball at an angle of with initial velocity .
(a) Estimate if the ball is hit the same velocity but the angle increased by .
(b) Estimate if the ball is hit the same angle but the velocity increased by 4 .
(Give your answer to four decimal places.)
(a)
(b)

Solución de inteligencia artificial de Upstudy

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Solución

Beyond the Answer

Part (a): Increase the angle by

Part (b): Increase the velocity by

Numerical Calculations

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A golfer hits a golf ball at an angle of with initial velocity . (a) Estimate if the ball is hit the same velocity but the angle increased by . (b) Estimate if the ball is hit the same angle but the velocity increased by 4 . (Give your answer to four decimal places.) (a) (b)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

Part (a): Increase the angle by

Part (b): Increase the velocity by

Numerical Calculations

preguntas relacionadas

Latest Calculus Questions

A golfer hits a golf ball at an angle of with initial velocity .
(a) Estimate if the ball is hit the same velocity but the angle increased by .
(b) Estimate if the ball is hit the same angle but the velocity increased by 4 .
(Give your answer to four decimal places.)
(a)
(b)