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 Flynn Franklin. 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 Flynn Franklin. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

To solve this problem, we will use the equations of projectile motion to estimate the change in distance $$ \Delta s $$ for both scenarios.

### Step 1: Understanding the Problem
The range $$ R $$ of a projectile launched at an angle $$ \theta $$ with an initial velocity $$ v $$ is given by the formula:
$$
R = \frac{v^2 \sin(2\theta)}{g}
$$
where $$ g $$ is the acceleration due to gravity. For this problem, we will assume $$ g \approx 32.2 \, \text{ft/s}^2 $$.

### Step 2: Calculate the Initial Range
First, we will calculate the initial range $$ R_0 $$ using $$ \theta = 24^\circ $$ and $$ v = 110 \, \text{ft/s} $$.

### Step 3: Calculate the Range for the Increased Angle
Next, we will calculate the range $$ R_1 $$ when the angle is increased by $$ 3^\circ $$ (i.e., $$ \theta = 27^\circ $$).

### Step 4: Calculate the Range for the Increased Velocity
Then, we will calculate the range $$ R_2 $$ when the velocity is increased by $$ 4 \, \text{ft/s} $$ (i.e., $$ v = 114 \, \text{ft/s} $$) while keeping the angle the same.

### Step 5: Calculate $$ \Delta s $$
Finally, we will find $$ \Delta s $$ for both cases:
- $$ \Delta s_1 = R_1 - R_0 $$
- $$ \Delta s_2 = R_2 - R_0 $$

Let's perform these calculations.

1. Calculate $$ R_0 $$
2. Calculate $$ R_1 $$
3. Calculate $$ R_2 $$

Now, I will compute these values.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(110^{2}\sin\left(\frac{2\times 27\pi }{180}\right)\right)}{32.2}$$
- step1: Remove the parentheses:
  $$\frac{110^{2}\sin\left(\frac{2\times 27\pi }{180}\right)}{32.2}$$
- step2: Reduce the fraction:
  $$\frac{110^{2}\sin\left(\frac{3\pi }{10}\right)}{32.2}$$
- step3: Evaluate the power:
  $$\frac{12100\sin\left(\frac{3\pi }{10}\right)}{32.2}$$
- step4: Convert the expressions:
  $$\frac{12100\sin\left(\frac{3\pi }{10}\right)}{\frac{161}{5}}$$
- step5: Multiply by the reciprocal:
  $$12100\sin\left(\frac{3\pi }{10}\right)\times \frac{5}{161}$$
- step6: Multiply:
  $$\frac{12100\sin\left(\frac{3\pi }{10}\right)\times 5}{161}$$
- step7: Multiply:
  $$\frac{60500\sin\left(\frac{3\pi }{10}\right)}{161}$$
- step8: Calculate the trigonometric value:
  $$304.009492$$
Calculate or simplify the expression $$ (114^2 * \sin(2 * 24 * \pi / 180)) / 32.2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(114^{2}\sin\left(\frac{2\times 24\pi }{180}\right)\right)}{32.2}$$
- step1: Remove the parentheses:
  $$\frac{114^{2}\sin\left(\frac{2\times 24\pi }{180}\right)}{32.2}$$
- step2: Reduce the fraction:
  $$\frac{114^{2}\sin\left(\frac{4\pi }{15}\right)}{32.2}$$
- step3: Evaluate the power:
  $$\frac{12996\sin\left(\frac{4\pi }{15}\right)}{32.2}$$
- step4: Convert the expressions:
  $$\frac{12996\sin\left(\frac{4\pi }{15}\right)}{\frac{161}{5}}$$
- step5: Multiply by the reciprocal:
  $$12996\sin\left(\frac{4\pi }{15}\right)\times \frac{5}{161}$$
- step6: Multiply:
  $$\frac{12996\sin\left(\frac{4\pi }{15}\right)\times 5}{161}$$
- step7: Multiply:
  $$\frac{64980\sin\left(\frac{4\pi }{15}\right)}{161}$$
- step8: Calculate the trigonometric value:
  $$299.935098$$
Calculate or simplify the expression $$ (110^2 * \sin(2 * 24 * \pi / 180)) / 32.2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(110^{2}\sin\left(\frac{2\times 24\pi }{180}\right)\right)}{32.2}$$
- step1: Remove the parentheses:
  $$\frac{110^{2}\sin\left(\frac{2\times 24\pi }{180}\right)}{32.2}$$
- step2: Reduce the fraction:
  $$\frac{110^{2}\sin\left(\frac{4\pi }{15}\right)}{32.2}$$
- step3: Evaluate the power:
  $$\frac{12100\sin\left(\frac{4\pi }{15}\right)}{32.2}$$
- step4: Convert the expressions:
  $$\frac{12100\sin\left(\frac{4\pi }{15}\right)}{\frac{161}{5}}$$
- step5: Multiply by the reciprocal:
  $$12100\sin\left(\frac{4\pi }{15}\right)\times \frac{5}{161}$$
- step6: Multiply:
  $$\frac{12100\sin\left(\frac{4\pi }{15}\right)\times 5}{161}$$
- step7: Multiply:
  $$\frac{60500\sin\left(\frac{4\pi }{15}\right)}{161}$$
- step8: Calculate the trigonometric value:
  $$279.256285$$
The calculations yield the following results:

1. The initial range $$ R_0 $$ when $$ \theta = 24^\circ $$ and $$ v = 110 \, \text{ft/s} $$:
   $$
   R_0 \approx 279.2563 \, \text{ft}
   $$

2. The range $$ R_1 $$ when the angle is increased to $$ 27^\circ $$:
   $$
   R_1 \approx 304.0095 \, \text{ft}
   $$

3. The range $$ R_2 $$ when the velocity is increased to $$ 114 \, \text{ft/s} $$:
   $$
   R_2 \approx 299.9351 \, \text{ft}
   $$

### Step 6: Calculate $$ \Delta s $$

Now, we can find $$ \Delta s $$ for both cases:

(a) For the increase in angle:
$$
\Delta s_1 = R_1 - R_0 = 304.0095 - 279.2563
$$

(b) For the increase in velocity:
$$
\Delta s_2 = R_2 - R_0 = 299.9351 - 279.2563
$$

Now, I will compute these differences.
Calculate the value by following steps:
- step0: Calculate:
  $$304.009492-279.256285$$
- step1: Subtract the numbers:
  $$24.753207$$
Calculate or simplify the expression $$ 299.935098 - 279.256285 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$299.935098-279.256285$$
- step1: Subtract the numbers:
  $$20.678813$$
The calculations for $$ \Delta s $$ yield the following results:

(a) The change in distance when the angle is increased by $$ 3^\circ $$:
$$
\Delta s_1 = 24.7532 \, \text{ft}
$$

(b) The change in distance when the velocity is increased by $$ 4 \, \text{ft/s} $$:
$$
\Delta s_2 = 20.6788 \, \text{ft}
$$

### Final Answers
- (a) $$ \Delta s \approx 24.7532 $$
- (b) $$ \Delta s \approx 20.6788 $$

These results are rounded to four decimal places as requested.
(a) $$ \Delta s = 24.7532 $$ (b) $$ \Delta s = 20.6788 $$

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)

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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)

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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)