James Bond is trying to jump from a bridge and land on the first car of a moving train that is going to pass under the bridge. The train moves at 85 km/h and the vertical height between the bridge and the top of the carriage is 15 m. if Bond gets a running start of 4.2 m/s horizontally as he leaps off the bridge, where would the train be ( relative to the bridge ) when he jumps off?
To determine where the train would be relative to the bridge when James Bond jumps off, we need to calculate the horizontal distance the train will cover during the time it takes for Bond to fall from the bridge and reach the height of the train's carriage.
First, let's calculate the time it takes for Bond to reach the height of the train's carriage. We can use the formula:
Δy = Viy * t + (1/2) * ay * t^2
where:
Δy = total vertical distance (15 m)
Viy = initial vertical velocity (0 m/s, as Bond starts from the bridge level)
ay = acceleration due to gravity (-9.8 m/s^2, taken as negative since it acts downward)
t = time
Rearranging the equation to solve for time:
15 m = 0 * t + (1/2) * (-9.8 m/s^2) * t^2
15 m = (-4.9 m/s^2) * t^2
t^2 = - (15 m) / (-4.9 m/s^2)
t^2 = 3.0612 s^2
t ≈ √3.0612
t ≈ 1.7493 s
Now that we have the time it takes for Bond to reach the height of the train's carriage, we can calculate the horizontal distance the train will cover in that time. Since the train's speed is given in km/h, we need to convert it to m/s:
85 km/h = 85,000 m/3600 s ≈ 23.6 m/s
Using the formula:
Δx = Vx * t
where:
Δx = horizontal distance
Vx = horizontal velocity (4.2 m/s, as given)
t = time (1.7493 s, as calculated above)
Δx = 4.2 m/s * 1.7493 s
Δx ≈ 7.3557 m
Therefore, when James Bond jumps off the bridge, the train would be approximately 7.4 meters ahead (relative to the bridge).
To determine where the train would be relative to the bridge when James Bond jumps off, we need to find the horizontal distance the train would have traveled during his jump.
First, we need to determine the time it takes for Bond to reach the train. We can use the formula for horizontal motion:
distance = speed × time
Bond's horizontal speed is 4.2 m/s, so we can rearrange the formula to solve for time:
time = distance / speed
Given that Bond's horizontal distance is 4.2 m and his speed is 4.2 m/s, we have:
time = 4.2 m / 4.2 m/s = 1 second
This means it will take Bond 1 second to reach the train.
Now, we can calculate the horizontal distance the train would have traveled during that 1-second interval. The formula for horizontal motion is:
distance = speed × time
The train's speed is 85 km/h, which can be converted to meters per second:
85 km/h × (1000 m/1 km) × (1 h/3600 s) = (85 × 1000) / 3600 = 23.61111 m/s (rounded to 5 decimal places)
Using this speed and the time of 1 second, we can calculate the distance:
distance = 23.61111 m/s × 1 s = 23.61111 m (rounded to 5 decimal places)
Therefore, the train would have traveled approximately 23.61111 meters relative to the bridge when Bond jumps off.