A football player kicks a ball with an initial velocity of 25 m/s at an angle of 53° above the horizontal. What will be the vertical component of the initial velocity of the ball? Round off your answer to 2 decimal places.
A football is kicked at an angle of 25 degrees and travels 50 yards. If the football is kicked at the same speed, what other angle will allow the football to travel 50 yards?
A 12 kg object has a momentum of 20 kg∙m/s toward the east. What is the object’s speed? Round off to 2 decimal places.
To find the vertical component of the initial velocity of the ball, we need to determine the vertical component of the overall velocity. We can use trigonometry to solve this.
Step 1: Given that the initial velocity (v0) is 25 m/s and the angle (θ) is 53°.
Step 2: The vertical component (v0y) can be found using the formula: v0y = v0 * sin(θ).
Step 3: Substitute the given values into the formula: v0y = 25 * sin(53°).
Step 4: Use a scientific calculator to evaluate the sine of 53°: sin(53°) ≈ 0.7986.
Step 5: Calculate the vertical component: v0y ≈ 25 * 0.7986 ≈ 19.97 m/s.
Therefore, the vertical component of the initial velocity of the ball is approximately 19.97 m/s.
In the next question, we are given that the football is kicked at an angle of 25 degrees and travels 50 yards. We need to find the angle that will allow the football to travel the same distance if kicked at the same speed.
Step 1: Given that the angle (θ) is 25° and the distance (d) is 50 yards.
Step 2: We can use the same formula as above to find the horizontal component of the velocity (v0x), as the speed is the same in both cases: v0x = v0 * cos(θ).
Step 3: The distance traveled is given by the formula: d = v0x * t. However, we want to find the angle, so we need to consider both the horizontal and vertical components.
Step 4: When a projectile reaches the same height on its projectile path, the time of flight is the same. So we can equate the time of flight for both cases and solve for the unknown angle.
Step 5: The time of flight is given by the formula: t = 2 * v0y / g, where g is the acceleration due to gravity.
Step 6: Substitute the given values of v0y and g into the formula: t = 2 * 19.97 m/s / 9.8 m/s² ≈ 4.07 seconds.
Step 7: Rearrange the formula for distance to solve for the horizontal component of the velocity: v0x = d / t.
Step 8: Convert the distance traveled from yards to meters (1 yard = 0.9144 meters): d = 50 yards * 0.9144 meters/yard ≈ 45.72 meters.
Step 9: Substitute the values of the distance and time into the formula: v0x = 45.72 meters / 4.07 seconds ≈ 11.23 m/s.
Step 10: Use the formula v0x = v0 * cos(θ) and the value of v0x to find the cosine of the unknown angle: cos(θ) = v0x / v0.
Step 11: Substitute the values of v0x and v0 into the formula: cos(θ) = 11.23 m/s / 25 m/s ≈ 0.4492.
Step 12: Use the inverse cosine function on a scientific calculator to find the unknown angle: θ ≈ cos^(-1)(0.4492) ≈ 63.52°.
Therefore, the other angle that would allow the football to travel 50 yards when kicked at the same speed is approximately 63.52°.
Moving on to the third question, we are given that an object with a mass of 12 kg has a momentum of 20 kg∙m/s toward the east. We need to find the object's speed.
Step 1: Given that the momentum (p) is 20 kg∙m/s and the mass (m) is 12 kg.
Step 2: The momentum is defined as the product of the mass and velocity: p = m * v.
Step 3: Rearrange the formula to solve for the velocity: v = p / m.
Step 4: Substitute the given values into the formula: v = 20 kg∙m/s / 12 kg.
Step 5: Calculate the velocity: v ≈ 1.67 m/s.
Therefore, the object's speed is approximately 1.67 m/s.
To find the vertical component of the initial velocity of the ball, we need to use trigonometry. Since the angle is given as 53° above the horizontal, we can use the sine function to find the vertical component. Here's how you can calculate it:
1. Convert the angle to radians: π/180 * 53° = 0.925 radians.
2. Use the sine function to find the vertical component: sin(0.925) = 0.800.
3. Multiply the result by the initial velocity: 0.800 * 25 m/s = 20 m/s.
So, the vertical component of the initial velocity of the ball is 20 m/s.
For the next question, we are given that the football is kicked at an angle of 25 degrees and travels 50 yards. We need to find the other angle that allows the football to travel the same distance of 50 yards when kicked at the same speed. Here's how you can determine it:
1. Using the same speed, we know that the initial velocity will be the same.
2. The horizontal component of the initial velocity will remain unchanged because it depends on the initial speed.
3. Therefore, the vertical component must be different to account for the change in distance.
4. We know that the horizontal distance traveled can be calculated as: horizontal distance = initial velocity * time * cosine(angle).
5. Since we want the horizontal distance to remain the same, we can equate the two distances as: initial horizontal distance = new horizontal distance.
6. Plug in the given values: 50 yards = 50 yards.
7. Use the formula for horizontal distance and simplify the equation: initial velocity * time * cosine(25°) = initial velocity * time * cosine(new angle).
8. Simplify further: cosine(25°) = cosine(new angle).
9. To find the new angle, use the inverse cosine function: new angle = arccos(cosine(25°)).
- Since the cosine function is periodic, we know that arccos(cosine(25°)) will give us multiple angles that satisfy the equation. In this case, the angles will be in the second and third quadrants, so we need to take the negative of the angle in the third quadrant.
10. Calculate the new angle: new angle = -arccos(cosine(25°)).
So, to allow the football to travel 50 yards when kicked at the same speed, the new angle would be -arccos(cosine(25°)).
For the last question, we are given that a 12 kg object has a momentum of 20 kg·m/s toward the east, and we need to find the object's speed. Here's how you can calculate it:
1. The momentum of an object is defined as the product of its mass and velocity: momentum = mass * velocity.
2. Rearrange the equation to solve for velocity: velocity = momentum / mass.
3. Plug in the given values: velocity = 20 kg·m/s / 12 kg.
4. Calculate the velocity: velocity ≈ 1.67 m/s.
Therefore, the object's speed is approximately 1.67 m/s toward the east.